Matholia Mathematics Primary 6A

M AT H E M AT I C S Workt ext

6A

for learners 11 - 12 years old

Aligned to the US Common Core State Standards

Matholia Mathematics Matholia Mathematics is a series covering levels K-6 and is fully aligned to the United States Common Core State Standards (USCCSS). Each level consists of two books (Book A and Book B) and combines textbook-style presentation of concepts as well as workbook practice.

Central to the USCCSS is the promotion of problem-solving skills and reasoning.

Matholia Mathematics achieves this by teaching and presenting concepts through a problem-solving based pedagogy and using the concrete-pictorial-abstract (CPA) approach. Learners acquire knowledge and understanding of concepts through a guided progression beginning with concrete examples and experiences which then flow into pictorial representations and finally mastery at the abstract and symbolic level. This approach ensures that learners develop a fundamental understanding of concepts rather than answering questions by learned procedures and algorithms.

Key features of the series include:

Anchor Task

1

3

Integers

Anchor Task

Open-ended activities serve as the starting point for understanding new concepts. Learners engage in activities and discussions to form concrete experiences before the concept is formalized.

Fractions

Anchor Task

o ChicFinae g

Cheesecake Recipe

–8ºC h: 4º

2º Low: –1

Serves 10 people Prep Time Cook Time Cooling Time

Hig

Cheesecake

–12

4

y Monda y Tuesda

0

8

sday Wedne ay

–2

–14

0

–7

14 Graham Crackers 1

1 cup pecans 2

4 tbsp butter 1 cup sugar 4 1 tsp cinnamon 2

3 1 tbsp flour 2

5 eggs 2 egg yolks

Friday

98

2

Let’s Learn

Crust

2 1 lbs cream cheese 2 3 cup sour cream 4 1 tsp salt 2 3 1 cups sugar 4

–2

12

Thursd

Concepts are presented in a clear and colorful manner. Worked problems provide learners with guided step-by-step progression through examples. Series mascots provide guidance through helpful comments and observations when new concepts are introduced.

30 minutes 1 hour 15 minutes

Ingredients

Let’s Learn

When findi ng the area of triangles, height. we first

need to find

Lets find the

the base and

base and heig

hts of som

(a)

A

e more trian

gles.

N

The height must be perpendic ular to the base.

In this triangle, the height is not a side length.

height

B

M

C We can choo se any side of the trian let's take the gle to be the base to be base. For the BC. triangle ABC, The height of the trian gle is given chosen base by the perpendi . This is a right cular heig height.

-angled trian ht to our gle, so AB is the perpendi cular BC is the base and AB is the height.

P

(b)

base

O

If we choose the base to be MO, the is perpendi cular to the height is given base. by the line NP

which

T

A

height height S base B base

26

C

For a base SU, the perp endicular heig triangle at line TV. ht is

U

located outs ide

V

the

27

ii

Let’s Practice

D

(d)

Let’s Practice

1.

height of the Identify the base and

height =

B

(a)

Learners demonstrate their understanding of concepts through a range of exercises and problems to be completed in a classroom environment.

Questions provide a varying degree of guidance and scaffolding as learners progress to mastery of the concepts.

base =

triangles.

base = height = F

G

E

C

A

F

(e) Q

(b)

base = base =

height =

height =

W

(f)

M

I

H

G

R

S

P

(c)

base = height =

base = height =

Z X

32

33

At Home

1.

Complete the followin g. Show your workin in its simplest form. g and write your answer (a)

4 x 7 5

(c)

2.

Multiply the mixed numbers. Show your working and write answer in its simples your t form.

(a) 3 2 x 5 5

(b) 6 x 2 3

At Home Further practice designed to be completed without the guidance of a teacher. Exercises and problems in this section follow on from those completed under Let’s Practice.

Y

P

O

N

5 x 8 12

(d) 10 x 5 6

(b) 2 x 3 5 8

(c)

(e) 6 x 3 7

(f)

4x22 3

(d) 7 x 3 3 5

7 x 9 5 (e) 12 x 4 1 8

(f)

8x53 12

1 12

1 13

Hands On Learners are encouraged to ‘learn by doing’ through the use of group activities and the use of mathematical manipulatives.

with the of a rectangle Hands On half the area a triangle is the area of Show that cm. and height. height of 12 same base a and 16 cm a width of below has The rectangle of 192 cm2 . height of 12 cm and a It has an area a base of 16 site page has on the oppo The triangle d lines so the cm. along the dotte neatly fill up to . Then cut w page belo the grid triangle from nge the pieces in the Cut out the 3 pieces. Arra triangle is in rectangle. half of the

Solve It! 1.

The figures are made up of semicircles (half circles) and straight lines.

Can you find the area of each figure? Take π = 3.14 and round off your answer to 1 decimal place.

39

(a)

Solve It!

2m

38

2m

Activities that require learners to apply logical reasoning and problem-solving. Problems are often posed which do not have a routine strategy for solving them. Learners are encouraged to think creatively and apply a range of problem-solving heuristics.

(b) 2 cm

2 cm 5 cm

2.

The figure below is made from a square of side length 6 cm.

The circular hole in the middle has a diamete r that is 2 the side length of the square. Find the 3 area of the figure. Take π = 3.14 and round off your answer to 1 decima l place.

21

Looking Back 1.

Looking Back

Express the percentage as a decimal

4.

(a) 12%

What percentage of each square is

colored?

(a)

(b)

(c)

Consolidated practice where learners demonstrate their understanding on a range of concepts taught within a unit.

(c)

(e) 70%

(d)

5.

2.

Color 14% of the square.

3.

42%

and fraction in its simplest form. (b) 28%

(d) 86%

(f)

50%

percentage.

Express the fraction as a decimal and 4 17 (b) 20 (a) 100

Color 45% of the square.

(c)

15 60

(d)

66 88

235 234

iii

Contents 1

iv

Integers Understanding Integers Comparing and Ordering Integers Operations on Integers Word Problems

2 3 12 17 27

2 Algebra Algebraic Expressions Evaluating Algebraic Expressions Simplifying Algebraic Expressions Solving Algebraic Expressions Word Problems

66 38 38 56 66 76 84

3 Fractions Multiplying Fractions Fractions and Division Word Problems

98 99 116 126

4 Ratio Ratio and Fraction Ratio and Proportion Word Problems

138 138 168 184

5 Percentage What Is Percentage?

Finding Percentage Percentage Increase and Decrease

194 194 204 218

6 Mid-year Exam Section A Section B Section C

240 240 248 255

v

1

Integers

Anchor Task

o g a c i h C Fine

–8ºC h: 4º

2º H 1 – : w o L

y Tuesda ay

sd Wedne

y Thursda

2

–12

4

y Monda

Friday

ig

12

–2

8

0

–2

–14

0

–7

Understanding Integers Let’s Learn Integers are whole numbers. The numbers on this number line are integers.

0

1

2

3

4

5

6

7

Are decimals and fractions integers?

8

The number line continues in both directions from 0. Numbers to the left of 0 are less than 0. Integers that are less than 0 are negative integers.

Integers that are greater than 0 are called positive integers.

–5

–4

–3

–2

–1

0

1

2

3

4

5

We use a minus sign (–) to show an integer is negative.

We write: –5 We say: negative five

0 is an integer that is neither positive nor negative. negative integers

positive integers 0

3

When the temperature falls below 0ºC, we read the temperature as a negative number. ºC

ºC

30

30

20

20

10

10

0

0

–10

–10

–20

–20

The temperature is 5ºC.

The temperature has fallen by 10ºC. It is now –5ºC.

The temperature is –5ºC. 10 units

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

Compare the thermometers. How has the temperature changed? ºC

ºC

30

30

20

20

10

10

0

0

–10

–10

–20

–20

The temperature increased by 5ºC. It is now 0ºC. 4

0ºC is 5ºC warmer than –5ºC. –5ºC is 5ºC cooler than 0ºC.

Pairs of integers that are the same distance from 0 are called opposites.

5 units

5 units 2 units

–5

–4

–3

–2

–1

2 units 0

1

2

3

4

5

0 is its own opposite.

–2 and 2 are opposites –5 and 5 are opposites

We can find the opposite of an integer by writing a minus sign in front of it.

Integer 5 –4 12 –1 24

Opposite

The negative of a negative is a positive!

–5 –(–4) = 4 –12 –(–1) = 1 –24

Consider the opposites –4 and 4.

4 units –5

–4

–3

–2

4 units –1

0

1

2

3

4

5

These integers are both an equal distance of 4 units from 0.

An integer's distance from 0 is expressed as its absolute value.

Both –4 and 4 have an absolute value of 4.

We write: |–4| We say: The absolute value of negative 4.

5

Let’s Practice 1. Write the integer indicated on the number line. Find its opposite. (a)

opposite:

–4

(b)

–2

0

2

4

–10

0

10

20

–2

0

2

4

0

20

40

opposite:

–40

6

4

opposite:

–4

(e)

2

opposite:

–20

(d)

0

opposite:

–4

(c)

–2

–20

2. Fill in the blanks. (a)

units –6

–5

–4

–3

–2

–1

0

units to the right of 1 is

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

.

(b)

units –6

–5

–4

–3

–2

–1

0

units to the left of 1 is

(c)

.

units –6

–5

–4

–3

–2

–1

units to the right of

(d)

0

is

.

units –6

–5

–4

–3

–2

–1

units to the left of

(e)

0

1

is

.

units –6

1

–5

–4

–3

–2

units to the right of

–1

0

is

1

.

7

3. Find the opposite and absolute value of each integer. (a) 3 opposite:

absolute value:

(b) –4 opposite: (c)

absolute value:

5

opposite:

absolute value:

(d) –8 opposite:

absolute value:

4. Read and answer the questions. (a) The temperature was 2ºC. It increased by 5ºC. What is the temperature now? (b) The temperature changed from –6ºC to 3ºC. How much did the temperature rise? (c) The temperature was 0ºC and fell 12ºC. What is the temperature now? (d) The temperature is 4ºC. How much cooler is –3ºC?

8

Solve It! Use the diagram to answer the questions.

4m

3m

2m 4m

12 m

7m

(a) How deep is the ocean?

(b) How much further does the diver need to dive to reach the treasure chest?

(c) The seagull sees the fish and dives into the ocean to catch it.

How far did the fish dive through the air and water?

9

At Home 1. Read and circle the correct integer on the number line. (a) 2 units to the right of 1.

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–2

–1

0

1

2

3

4

5

6

–2

–1

0

1

2

3

4

5

6

–2

–1

0

1

2

3

4

5

6

–1

0

1

2

3

4

5

6

–1

0

1

2

3

4

5

6

(b) 3 units to the left of 1.

–6

(c)

–5

–4

–3

The opposite of 5.

–6

–5

–4

–3

(d) The opposite of –4.

–6

–5

–4

–3

(e) 4 units to the right of –3.

–6

–5

–4

–3

–2

(f) 5 units to the left of 2.

–6

10

–5

–4

–3

–2

2. Find the opposite and absolute value of each integer. (a) 2 opposite:

absolute value:

(b) –3 opposite: (c)

absolute value:

0

opposite:

absolute value:

(d) –12 opposite:

absolute value:

3. Color to show the temperature. (a) 4ºC warmer than 0ºC. (b) 10ºC cooler than 8ºC. ºC

ºC

30

30

20

20

10

10

0

0

–10

–10

–20

–20

11

Comparing and Ordering Integers Let’s Learn Chelsea measured the temperature outside 4 times during the day.

She recorded her observations in the table.

Time Temperature

08 00 –2ºC

12 00 3ºC

16 00 –5ºC

20 00 –8ºC

To compare the temperatures, we can place them on a number line.

20 00

–10

–9

–8

16 00

–7

–6

–5

08 00

–4

–3

–2

12 00

–1

0

1

2

3

4

(a) At what time was the temperature the lowest?

–8 is the furthest number to the left. It is the lowest number.

The temperature was lowest at 20 00. (b) At what time was the temperature the highest?

3 is the furthest number to the right. It is the highest number.

The temperature was the highest at 12 00. (c)

Compare the temperatures at 08 00 and 12 00.

–2 is more left than 3 on the number line. It is the lower number.

We can write: –2ºC < 3ºC and 3ºC > –2ºC (d) Arrange the temperatures from the highest to the lowest.

Let's write each temperature as they appear from right to left on the number line.

3ºC, –2ºC, –5ºC and –8ºC. 12

5

Let’s Practice 1. Circle the numbers on the number line. Fill in the blanks to compare. (a) 3 and 0 –6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Which number is on the left?

Which number is smaller?

<

(b) –2 and 1 –6

–5

–4

–3

–2

–1

Which number is on the left?

Which number is smaller?

(c)

< 5 and –3 –6

–5

–4

–3

–2

–1

Which number is on the right?

Which number is greater?

>

(d) |–2| and –6 –6

–5

–4

–3

–2

–1

Which number is on the right?

Which number is greater?

> 13

2. Write the numbers on the number line. Then arrange them from the smallest to the greatest. (a) -1, 5 and 1 0

<

<

(b) |–3|, 4 and -4 0

<

<

3. Compare the numbers by writing >, < or = symbols. (a) –4

0 (b) –3

(d) 1

8 (e) –4

3 (c)

–2 (f) 5

4. Arrange the numbers from the greatest to the smallest. (a) 1, 3, 0 (b) 5, |–2|, –3

(c)

>

>

>

>

|-1|, -1, 0 (d) 5, –2, –3 >

>

>

>

5. Arrange the numbers from the smallest to the greatest. (a) 0, –5, –10, 1

<

<

<

<

<

(b) 2, –2, –3, |–15|

14

<

–6

–8 5

Solve It! The map below shows the temperatures across Europe on a winter's day.

–2ºC –1ºC

–4ºC

0ºC

–3ºC

–2ºC 3ºC –1ºC 7ºC 4ºC

3ºC

5ºC

(a) Which city has the lowest temperature? (b) Which city has the highest temperature? (c)

Name a pair of cities that recorded the same temperature.

and

(d) Name a pair of cities that recorded opposite temperatures.

and

(e) Which cities recorded the 4 lowest temperatures?

,

,

and

15

At Home 1. Write the numbers on the number line.

Then arrange them from the greatest to the smallest (a) –3, –4 and 0 0

>

>

(b) 2, 5 and -5 0

>

>

2. Circle the numbers smaller than –3.

Cross the numbers that are greater than 3.

–4

0 –1

4

6 –9

–3

–5

3. Arrange the numbers from the greatest to the smallest. (a) –4, –1, –10, 0

>

>

>

>

>

(b) 5, –12, 9, –2

16

>

1

–12 2 5

Operations on Integers Let’s Learn We can use a number line to show addition and subtraction of integers.

The number line below shows 3 + 2 = 5.

2 units

3 units –1

0

1

When adding, we move right along the number line.

2

3

4

5

6

The number line below shows –5 + 3 = –2.

3 units 5 units –6

–5

–4

–3

–2

–1

0

1

The number line below shows 3 + (–6) = –3.

6 units

–6 units to the left of the number line.

3 units –4

–3

–2

–1

0

1

2

3

4

3 + (–6) = 3 – 6 = –3 Adding a negative integer is the same as subtracting a positive integer.

17

The number line below shows –2 – 3 = –5.

3 units 2 units –6

–5

–4

–3

–2

–1

0

1

–2 – 3 = –2 + (–3) = –5 Subtracting an integer is the same as adding its opposite.

The number line below shows –2 – (–6) = 4.

6 units 2 units –3

–2

–1

0

1

2

3

4

Subtracting –6 is the same as adding its opposite, 6.

–2 – (–6) = –2 + 6 = 4

The product of integers of the same sign is positive.

You are familiar with the product of positive integers.

Let's look at some examples.

3 x 4 = 12 5 x 2 = 10 16 x 3 = 48 The product of negative integers is also positive.

Let's look at some examples.

–3 x (–4) = 12 –6 x (–5) = 30 –20 x (–8) = 160

18

We add the opposite of –6, which is 6.

The product of integers of different signs is negative.

Let's look at the products of integers with different signs.

–3 x 6 = –18 7 x (–3) = –21 –9 x 4 = –36 The quotient of integers of the same sign is positive.

You are familiar with the quotient of positive integers.

Let's look at some examples.

16 ÷ 2 = 8 18 ÷ 6 = 3 20 ÷ 4 = 5 The quotient of negative integers is also positive.

Let's look at some examples.

–16 ÷ (–2) = 8 –6 ÷ (–2) = 3 –28 ÷ (–7) = 4

The quotient of integers of different signs is negative.

Let's look at the quotients of integers with different signs.

–10 ÷ 5 = –2 12 ÷ (–3) = –4 –9 ÷ 3 = –3

19

Let’s Practice 1. Write an addition equation to match the number line. (a)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

(b)

(c)

(d)

(e)

(f)

20

2. Draw an arrow on the number line to represent the addition of integers.

Complete the addition equation. (a) –3 + 2 =

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

(b) –4 + 6 =

–6

(c)

–1 + 5 =

–6

(d) 2 + (–4) =

–6

–5

(e) –5 + 3 =

–6

–5

(f) 4 + (–7) =

–6

–5

21

3. Write a subtraction equation to match the number line. (a)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

(b)

(c)

(d)

(e)

(f)

22

4. Draw an arrow on the number line to represent the subtraction of integers. Complete the subtraction equation. (a) 1 – 5 =

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

(b) –2 – 3 =

–6

(c)

3–6=

–6

(d) –3 – (–4) =

–6

(e) 5 – 6 =

–6

(f) –4 – (–7) =

–6

–5

23

5. Add or subtract the integers. (a) –3 + 10 =

(b) 8 – 9 =

(c)

2–5=

(d) 4 + (–8) =

(e) –15 + 8 =

(f) –9 – (–6) =

(g) 20 – 22 =

(h) 16 – 20 =

(i) –11 + 11 =

(j) –4 + 19 =

(k) 6 – (– 6) =

(m) 0 – 10 =

(n) –3 – 0 =

(p) 1 + (–11) =

(q) –50 + 49 =

(l) 7 + (–6) = (o) 16 – (– 5) = (r) –2 + (– 2) =

6. Find the products. (a) 5 x 4 =

(b) –2 x 3 =

(d) –2 x 5 = (g) 4 x 6 =

(e) –7 x (–5) = (h) –1 x (–2) =

(c)

7x5=

(f) 3 x (–3) = (i) –4 x (–8) =

(j) –10 x 2 =

(k) –12 x (–5) =

(l) 6 x (–9) =

(m) 3 x (–16) =

(n) –8 x (–8) =

(o) 100 x (–2) =

(p) 7 x (–7) =

(q) 3 x (–25) =

(r) –9 x (–9) =

7. Find the quotients. (a) 14 ÷ (–2) =

(b) –18 ÷ (–6) =

(d) –4 ÷ (–2) =

(e) 42 ÷ (–7) =

(f) –30 ÷ 15 =

(g) 16 ÷ (–4) =

(h) –81 ÷ (–9) =

(i) –28 ÷ 4 =

(j) –6 ÷ (–2) =

(k) 24 ÷ (–6) =

(l) –40 ÷ 20 =

(m) 15 ÷ (–5) =

(n) –72 ÷ 9 =

(p) –45 ÷ (–9) = 24

(q) 48 ÷ (–8) =

(c)

–20 ÷ 5 =

(o) –100 ÷ 10 = (r) –56 ÷ 7 =

At Home 1. Write an addition equation to match the number line. (a)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–12

–10

–8

–6

–4

–2

0

2

4

6

8

10

12

(b)

2. Draw an arrow on the number line to represent the addition of integers.

Complete the addition equation. (a) –2 + 6 =

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–8

–6

–4

–2

0

2

4

6

8

10

12

–30 –25 –20 –15 –10

–5

0

5

10

15

20

25

30

(b) –12 + 8 =

–12

(c)

–10

25 + (–35) =

25

3. Write a subtraction equation to match the number line. (a)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

(b)

4. Draw an arrow on the number line to represent the subtraction of integers. Complete the subtraction equation. (a) 1 – 7 =

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

(b) –3 – 3 =

–6

–5

5. Complete the equations. (a) 4 + (–1) =

(b) –12 + (–2) =

(d) 12 ÷ (–2) =

(e) –18 ÷ (–9) =

(c)

6 + (–12) =

(f) –27 ÷ 3 =

(g) 5 – 10 =

(h) –2 – (–2) =

(i) 12 – (–5) =

(j) –19 x 2 =

(k) –12 x (–3) =

(l) 5 x (–6) =

26

Word Problems Let’s Learn When Keira woke up in the morning, the temperature outside was –8ºC. By the time she arrived at school, the temperature was 2ºC. Find the increase in temperature.

ºC

ºC

30

30

20

20

10

10

0

0

–10

–10

–20

–20

4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9

To find the increase in temperature, we need to find the difference.

Let's subtract the lower integer from the higher integer.

2 – (–8) = 10 The temperature increased by 10ºC.

27

An office building has 8 levels above ground and 5 basement levels.

Mr. Sato took the elevator from his office on the 5th floor to the 4th basement level. How many levels down did Mr. Sato move?

8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5

Let's find the difference.

5 – (–4) = 9 Mr. Sato moved down 9 levels.

Mrs. Yi has $100 in her savings account. Her car repayments are deducted from her savings account at a rate of $40 per week. Find Mrs. Yi's account balance after 4 weeks of car repayments.

–80 –60 –40 –20

0

20

40

60

80

100 120

100 – (4 x 40) = 100 – 160 = –60 Mrs. Yi's account balance is –$60. Her account is $60 in debt.

28

A car consumes 9 liters of fuel per hour. Find the change in fuel after 5 hours.

We can represent the consumption of fuel as a negative number.

–9 x 5 = –45 The car will consume 45 liters of fuel in 5 hours.

Look at the weather map and answer the questions below.

–19ºC 0ºC –6ºC 7ºC –3ºC

(a) How much warmer is Baltimore compared to Charleston?

7 – (–3) = 10 It is 10ºC warmer in Baltimore than Charleston. (b) Which city is 6ºC cooler than New York?

0 – 6 = –6 Pittsburgh is 6ºC cooler than New York. (c) What is the temperature difference between the coolest and warmest cities?

7 – (–19) = 26 The temperature difference between the coolest and warmest cities is 26ºC. 29

Let’s Practice 1. Sophie is cooking in the kitchen. She takes some chicken nuggets from the freezer and puts them into the oven. The freezer is set to –16ºC and the oven is set to 180ºC. Find the difference in temperature.

2. Mr. Finch is exercising and dieting to lose weight. He loses 2 kg every month. Express Mr. Finch's change in weight after 8 months.

3. Ethan's basement is 4 meters below ground level. His tree house is 3 meters above the ground. Find the difference in height of Ethan's basement and tree house.

30

4. The graph shows the daily profits of a noodle shop from Monday to Friday.

Noodle Shop Profits 30 25 20 15

Profit ($)

10 5 0 –5 –10 –15 –20 Mon

Tue

Wed Day

Thur

Fri

(a) On which days was the noodle shop profitable? (b) On which day did the noodle shop make the greatest profit? (c) On which day did the noodle shop make the greatest loss? (d) How much profit did the noodle shop make from Monday to Friday?

31

At Home 1. Mercury freezes at around –39ºC. Blake checks his temperature with a mercury thermometer. He records a temperature of 37ºC. How many degrees Celsius above freezing is the temperature of the mercury?

2. A rocket burns 50 kg of fuel every second. Express the change in mass of the rocket after 1 minute.

3. Halle owes her sister $25. She repays her $17 and then borrows another $8. How much money does Halle owe her sister now?

32

4. The table shows the freezing and boiling temperatures of some common solvents.

Solvent Water Acetic acid Ethanol Benzene Chloroform

Freezing Point (ºC) Boiling Point (ºC) 0 100 17 118 –115 78 6 80 –64 61

(a) What is the difference in temperature between the freezing and boiling point of water? (a) What is the difference in temperature between the freezing and boiling point of chloroform? (c) Which solvent has the greatest difference in temperature between its freezing and boiling points? (d) Which solvent has the smallest difference in temperature between its freezing and boiling points?

33

Looking Back 1. Find the opposite and absolute value of each integer. (a) 7 opposite:

absolute value:

(b) –1 opposite: (c)

absolute value:

–13

opposite:

absolute value:

(d) 0 opposite:

absolute value:

2. Read and answer the questions. (a) The temperature was –2ºC. It increased by 8ºC. What is the temperature now? (b) The temperature changed from –12ºC to –8ºC. How much did the temperature rise? (c) The temperature was 1ºC and fell 15ºC. What is the temperature now? (d) The temperature is 5ºC. How much cooler is –13ºC?

34

3. Write the numbers on the number line. Then arrange them from the smallest to the greatest. (a) -3, 3 and 0 0

<

<

(b) |–6|, 1 and -4 0

<

<

4. Compare the numbers by writing >, < or = symbols. (a) –2

1 (b) –4

(d) 1

1 (e) |–8|

4 (c)

0

8 (f) –(–2)

–8 –2

5. Arrange the numbers from the greatest to the smallest. (a) –3, 1, 0 (b) –3, |–2|, |–3|

(c)

>

>

>

>

4, |-1|, –2 (d) 3, –1, –7 >

>

>

>

6. Arrange the numbers from the smallest to the greatest. (a) 2, –15, –3, |-10|

<

<

<

<

<

(b) 1, –5, –3, |–12|

<

35

7. Write a subtraction equation to match the number line. (a)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

(b)

8. Draw an arrow on the number line to represent the subtraction of integers. Complete the subtraction equation. (a) 1 – 7 =

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

(b) –3 – 3 =

–6

–5

9. Complete the equations. (a) 8 + (–2) =

(b) –15 + (–7) =

(c)

(d) 18 ÷ (–2) =

(e) –14 ÷ (–7) =

(f) –36 ÷ 9 =

(h) –4 – (–5) =

(i) 12 – (–1) =

(k) –10 x (–3) =

(l) 7 x (–6) =

(g) 5 – 8 = (j) –16 x 2 = 36

3 + (–12) =

10. The temperature inside an ice box is –3ºC. As the ice melts, the temperature increases by 2ºC per hour. Find the temperature of the ice box after 10 hours.

11. A motorcycle consumes 4 liters of fuel every hour.

Find the change in fuel after 4 hours.

12. At the South Pole, the minimum average temperature in October is –54ºC and the maximum average temperature is –48ºC. Find the difference in temperature.

37

2

Algebra

Algebraic Expressions Anchor Task

= 10 = 12 =7

38

Let’s Learn Riley and Sophie are collecting seashells.

Riley has 5 seashells. Sophie has some seashells in a bucket. How many seashells do they have altogether?

x

5

?

The number of seashells in the bucket is an unknown quantity. We can use the letter x to represent the number of seashells in Sophie's bucket.

Riley and Sophie have (5 + x) seashells.

5 + x is an algebraic expression in terms of x.

In an algebraic expression, letters are used to represent unknown quantities.

Riley finds 3 more seashells. She now has 8 seashells in all.

3+5+x=8+x x

8

(8 + x)

Riley and Sophie have (8 + x) seashells in all.

39

Let's use algebraic expressions to express the age of Riley and her siblings.

I am 3 years older than Riley.

I am 2 years younger than Riley.

Jessica

Riley

Joanne

Riley's age is unknown. We can express Riley's age as y.

We can say, Riley is y years old.

We know that Jessica is 3 years older than Riley.

We can express Jessica's age in terms of Riley's age.

Jessica is (y + 3) years old.

We know that Joanne is 2 years younger than Riley.

We can express Joanne's age in terms of Riley's age.

Joanne is (y – 2) years old.

(y + 3) years old

40

y years old

(y – 2) years old

Each brick has the same mass. The mass of each brick is unknown. Let's use z to represent the mass of 1 brick.

z

z

z

z

z

z

?

We can say that each brick has a mass of z kg.

There are 6 bricks in all. We can express the total mass of the bricks using an algebraic expression. z+z+z+z+z+z=6xz We write 6 x z as 6z.

The total mass of the bricks is 6z kg.

A ribbon has a length of t m. Wyatt cut the ribbon into 3 pieces of equal length. Express the length of 1 piece of the ribbon in terms of t. tm

?

The length of 1 piece of ribbon is t ÷ 3.

We write t ÷ 3 as

t .

3

41

Blake cooked a pizza. He cut the pizza into b slices. He ate 2 slices then divided the remaining slices equally among his 4 friends. How many slices of pizza did each friend receive?

b

?

(b – 2) ÷ 4 =

2 slices

(b – 2) 4

Each friend received

(b – 2) slices of pizza.

4

Mrs. Johnston has 5 green apples. She goes to the grocery store and buys 3 bags of red apples. Each bag contains c apples. How many apples does Mrs. Johnston have in all?

c

5

?

5 + c + c + c = 5 + 3c Mrs. Johnston has 5 + 3c apples in all.

42

Let’s Practice 1. Write an algebraic expression. (a) Chelsea has x strawberries. She gives 3 strawberries to Sophie.

How many strawberries does Chelsea have now?

(b) Ethan scored m points on his science quiz. Blake scored 5 points more than Ethan. What was Blake's score?

(c) Riley cycled for n kilometers on Saturday. She cycled 9 kilometers on Sunday. How far did Riley cycle on the weekend?

43

(d) A packet contains b candies. The candies are shared equally among 5 friends. How many candies does each friend get?

(e) Riley used c beads to make a bracelet. She made 12 such bracelets.

How many beads did she use in all?

(f) The total length of 12 identical rulers is d centimeters.

What is the length of 1 ruler?

44

2. Write an algebraic expression. (a) 10 is added to x.

(b) Multiply m by 7.

(c)

20 is divided by y.

(d) There are 12 groups of t.

(e) 88 is subtracted from z.

45

3. Write the algebraic expression. p

5

(a)

t

(b)

(c)

m

m

f

(d)

9

(e)

12

h

46

m

4. Choose your own letter to write an algebraic expression. (a) A large bottle of orange juice is poured equally into 4 cups.

How much orange juice is in each cup?

(b) Blake has a jar of marbles. Ethan gives him 100 more marbles.

How many marbles does Blake have now?

(c) Mrs. Jenkins withdraws some money from the bank and spends $425 on groceries. How much money does Mrs. Jenkins have now?

(d) 12 children raised the same amount of money in a school fundraiser. How much did they raise in total?

47

5. Write the algebraic expression. (a) The distance around a running track is d meters. Ethan walks 500 meters then runs 8 laps around the track. What is the total distance Ethan covered?

(b) A tub of popcorn costs $p. An ice-cream costs $2 more than a tub of popcorn. What is the cost of 2 tubs of popcorn and an ice-cream?

(c) Riley received $x as a gift. She kept $5 and gave the rest to her 3 sisters in equal amounts. How much did each sister get?

48

6. Write an algebraic expression. (a) 10 is subtracted from the product of x and 5.

(b) The sum of y and 5 is divided by 12.

(c)

19 is subtracted from the product of m and 10.

(d) 25 is added to 12 groups of t.

49

At Home 1. Write an algebraic expression. (a) A monkey has g bananas. It finds 6 more bananas. How many bananas does the monkey have now?

(b) In a soccer match, Jordan scored m goals. Wyatt scored 2 fewer goals than Jordan. How many goals did Wyatt score?

(c) Every day, Mr. Whyte drives q kilometers. How far does Mr. Whyte drive in 2 weeks?

50

(d) Keira has a piece of ribbon that is h meters in length. She cuts the ribbon into 12 equal lengths. What is the length of 1 piece of ribbon?

(e) Halle is r years old. Chelsea is 5 years younger than Halle.

How old is Chelsea?

(f) There are w players in a football team. There are 25 football teams.

How many players are there in all?

51

2. Write an algebraic expression. (a) 15 is multiplied by x.

(b) Multiply m by 11.

(c)

29 is added to a.

(d) There are g groups of 8.

(e) 14 is subtracted from u.

52

3. Write the algebraic expression. x

(a)

(b)

x

e

(c)

13

(d)

65

a

53

4. Write the algebraic expression. (a) An egg costs p¢. A 50% discount is applied when buying eggs in a 12-pack. What is the cost of a 12-pack of eggs?

(b) A computer program takes a number n and doubles it.

It then subtracts 6 from the result.

(c) A bottle contains d liters of water. The water is poured in equal amounts into 4 glasses. 3 liters of water remain in the bottle.

How much water is in each glass?

54

5. Write an algebraic expression. (a) 10 is added to the product of w and 50.

(b) The sum of d and 12 is divided by 12.

(c)

The product of a and 10 is added to 14.

(d) 5 is subtracted from the quotient of 12 and t.

55

Evaluating Algebraic Expressions Let’s Learn A car tire has a mass of x kg.

Find the mass of 4 tires in terms of x. x kg + x kg + x kg + x kg = 4x kg The mass of 4 tires is 4x kg.

Find the total mass of 4 car tires when x = 6.

When x = 6 4x = 4 x x = 4 x 6 = 24 The total mass of 4 car tires is 24 kg.

Let's evaluate these algebraic expressions!

Find 6a – 5 when a = 7.

When a = 7, 6a – 5 = 6 x 7 – 5 = 42 – 5 = 37 Find 24 + 5y when y = 4.

When y = 4, 24 + 5y = 25 + 5 x 4 = 24 + 20 = 44

56

Find

(15 – 2r) when r = 3.

3

When r = 3, (15 – 2r) (15 – 2 x 3) = 3 3 15 – 6 = 3 9 = =3 3

Let’s Practice 1. Find the value of each expression when w = 3. (a) 25 – 2w

(b) 10w + 22

(c)

15 – w 4

(d)

6w + 18 6

(e) 62 + 12w

57

2. Find the value of each expression when x = 12. (a) 21 + 2x

(b) 7x + 16

(c)

88 – 7x 2

(d)

6x + 33 5

(e) 52 + 12x

58

3. Complete the tables. (a)

Expression

Value of Expression when s = 2

Value of Expression when s = 7

Value of Expression when t = 10

Value of Expression when t = 15

20s 12s – 24 7s + 7 7 122 – 12s 10s + 14 2 (b)

Expression 7t + 100 12t – 24 6 + 6t 3 12t – 19 22t + 15 5

59

4. Evaluate the algebraic expressions when x = 12 and y = 8. (a) 5x – 4y + 5

(b) 62 – 3x – 2y

(c)

500 – 11x + y

(d) 9x – 12y

60

5. Evaluate the algebraic expressions when r = 3 and s = 15. (a) r + 5s

(b) 10s + 5r – 100

(c)

18 – r + 5s

(d) 2s – 8r + 13

61

At Home 1. Find the value of each expression when z = 9. (a) 25 – z

(b) 9z – 11

(c)

144 – z 9

(d)

16z – 12 12

(e) 21z + 2z

62

2. Find the value of each expression when d = 8. (a) 21 + 8d

(b) 2d + 96

(c)

98 – 7d 7

(d)

11d + 2 9

(e) 24d + 327

63

3. Complete the tables. (a)

Expression

Value of Expression when k = 12

Value of Expression when k = 9

Value of Expression when c = 6

Value of Expression when c = 10

8k +8 12k – 44 56 + 2k 2 204 – 12k 12k + 8 4 (b)

Expression 6c + 4 8 100c – 550 18 + 3c 6 122 + 12c 4c + 14 2

64

4. Evaluate the algebraic expressions when v = 9 and w = 12. (a) v + 3w

(b) 10w + 10v – 100

(c)

12w – 7v 9

(d)

4v + 3w v

65

Simplifying Algebraic Expressions Let’s Learn Riley, Sophie and Halle each have $x.

How much money do they have in all? $x

$x

$x

They have ($x + $x + $x) in all.

We can simplify ($x + $x + $x) as $3x.

Riley, Sophie and Halle have $3x in all.

Jordan has 3y computer games. Wyatt has 2y computer games.

How many computer games do they have in all?

3y

2y

They have (3y + 2y) computer games in all.

We can simplify (3y + 2y) as 5y.

Jordan and Wyatt have 5y computer games in all.

Simplify 4z + 6z – 2z.

4z + 6z – 2z = 10z – 2z = 8z

66

Simplify 4a + 8 + 2a + 4.

First, group the numbers together and group the unknowns together.

Then simplify.

4a + 2a = 6a 8 + 4 = 12

4a + 8 + 2a + 4 = 4a + 2a + 8 + 4 = 6a + 12

Simplify 8b + 6 – 3b + 5b – 2.

8b – 3b + 5b = 5b + 5b = 10b 6–2=4

First, group the numbers together and group the unknowns together.

Then simplify.

8b + 6 – 3b + 5b – 2 = 8b – 3b + 5b + 6 – 2 = 10b + 4 Simplify c + 2d + 4c + 12 – d – 9.

First, group the numbers together and the unknowns together.

Then simplify. c + 2d + 4c + 12 – d – 9 = c + 4c + 2d – d + 12 – 9 = 5c + d + 3 There are 2 different unknowns, c and d.

c + 4c = 5c 2d – d = d 12 – 9 = 3

67

Let’s Practice 1. Simplify. Show your working. (a) a + a + a

(b) 10x – x

(c)

y+y+y+y+y+y

(d) 3e + 13e

(e) 30g – 21g

68

2. Simplify. Show your working. (a) 2s + 3s – 7

(b) 2r + 3r + 38

(c)

10v – v – 6

(d) 19 + 2x + 4 – x

(e) 11 + 5p + 13 – 2p

69

3. Simplify. Show your working. (a) 5d + 3d – c – d – 3d

(b) 5m + 5n + 23m + n + 5m

(c)

50b – 14b + 60c – 10c – 30c + 6

(d) 2g – g + 6 + 5h –2h + 6

(e) 70 + 2i + 5i – 12j – 6j + 15

70

Solve It! 1. A number is represented by v. The number is multiplied by 3.

5 is subtracted from the result. The new result is divided by 5. (a) Write an algebraic expression in terms of v. (b) What is the value of the final expression when v = 5? (c) What is the value of the final expression when v = 50?

2. A number is represented by w. The number is divided by 9.

12 is added to the result. The new result is multiplied by 6. (a) Write an algebraic expression in terms of w. (b) What is the value of the final expression when w = 18? (c) What is the value of the final expression when w = 81?

71

At Home 1. Simplify. Show your working. (a) 12 x a

(b) 10p – p – p

(c)

d + d + d + d + d + d+ d + d

(d) 23h + 3h

(e) 30q – 20q – 5q

72

2. Simplify. Show your working. (a) 110 – 7e – 5e – 43

(b) 12y + 4 – 3y – 3

(c)

7b + 23b + 6 + 40

(d) 19 + 15x + 14 – 5x

(e) 12m – 3 – 2 - m + 6m

73

3. Simplify. (a) 15s + 3s – t – s – 13s

(b) 15m + 3n + 23m – 2n + 15m

(c)

50c – 14c + 23d – 10d – 10d + 6+ 1

(d) 8x – x + 6 x + 15h –12h + 9 + 4 +4

(e) 26 + 12i + 15i – 12j – 6j + 15i – 14

74

Solve It! 1. A number is represented by x. The number is multiplied by 5.

12 is subtracted from the number. The number is divided by 6. (a) Write an algebraic expression in terms of x. (b) What is the value of the final expression when x = 12? (c) What is the value of the final expression when x = 6?

2. A number is represented by y. The number is divided by 8.

12 is added to the number. The number is multiplied by 15. (a) Write an algebraic expression in terms of y. (b) What is the value of the final expression when y = 64? (c) What is the value of the final expression when y = 8?

75

Solving Algebraic Equations Let’s Learn Halle had $x. Her grandmother gave her $10 and she had $23 in all.

How much did Halle have at first? $x

$10

?

Do you know what value of x will make the equation true?

x + 10 = 23 x + 10 – 10 = 23 – 10 = 13 Halle had $13 at first.

When the values of the unknowns in an equation have been found, we say we have solved the equation.

A baker baked 48 bread rolls. He sold y bread rolls and had 15 bread rolls left. How many bread rolls did the baker sell?

48

15 y + 15 = 48 y + 15 – 15 = 48 – 15 y = 33

76

y

If you change 1 side of an equation, you must always change the other side in the same way!

We can also use the balance method to visualize the solving of algebraic equations. Let's represent the equation 2x + 5 = 11 on a balance.

1 1 1 1 1

x x

1 1 1 1 1 1

1 1 1 1 1

2x + 5 = 11

Remove 5 units from both sides.

1 1 1 1 1

1 1 1 1 1

x x

1

Divide each side into equal parts.

x

x

1 1 1

1 1 1

The parts on both sides are equal. x

1 1 1

Subtract 5 from both sides.

1 1 1 1 1

2x + 5 – 5 = 11 – 5 2x = 6

Divide both sides by 2.

2x 6 = 2 2

We have solved the equation. x=3

Solve the equation 2w + 8 = 14.

2w + 8 = 14 2w + 8 – 8 = 14 – 8 2w = 6 2w ÷ 2 = 6 ÷ 2 w=3

77

Let’s Practice Solve the equations. Show your working. (a) t + 16 = 30

(b) 14c + 10 = 52

(c)

12z – 6 = 42

(d) 4y – 20 = 100

(e) 5a – 7 + 2 = 5

78

(f) 2m – 100 = 0

(g) 4e – 8 = 20

(h) 36 – 2h = 10

(i) 54 – 9u = 9

(j) 5w + 100 = 125

79

At Home Solve the equations. Show your working. (a) 2b + 16 = 30

(b) 14t – 52 = 4

(c)

12p + 51 = 99

(d) 9w – 7 = 38

(e) 5x – 50 = 450

80

(f) 123 + 2m = 149

(g) 12e – 8 = 136

(h) 126 – 14h = 0

(i) 54 + 6 + 3u = 69

(j) 25v + 125 – 25 = 225

81

Solve It! 1. Sophie has a piece of red ribbon which is 245 cm in length. She has a piece of green ribbon which is x cm shorter than the red ribbon.

She has a piece of yellow ribbon which is 3 times longer than the green ribbon. (a) Express the length of the green ribbon in terms of x. (b) Express the length of the yellow ribbon in terms of x. (c) Find the total length of the green and yellow ribbons when x = 12.

82

2. Mrs. Brown earns $m per month. Mrs. Williams earns 4 times as much as Mrs. Brown. (a) Express Mrs. Brown's and Mrs. William's total income over a period of 6 months. (b) How much does each person earn over 1 year when m = $1,250?

83

Word Problems Let’s Learn The figure below shows the length of the fences of Mr. Rolland's paddock.

2x m

x + 18 m

xm 3x m

Express the perimeter of Mr. Rolland's paddock in terms of x.

Perimeter = 2x + x + 18 + x + 3x = 7x + 18 Find the perimeter of Mr. Rolland's paddock when x = 13.

When x = 13, Perimeter = = = =

7x + 18 7 x 13 + 18 91 + 18 109 m

When x = 13, the perimeter of Mr Rolland's paddock is 109 meters.

Find the perimeter of Mr. Rolland's paddock when x = 25.

When x = 25, Perimeter = 7x + 18 = 7 x 25 + 18 = 175 + 18 = 193 m When x = 25, the perimeter of Mr Rolland's paddock is 193 meters.

84

The table shows the number of pupils in Grades 1 to 6 attending Bright Sparks STEM School.

Student Enrollments – Bright Sparks STEM School Grade

Students Enrolled

1

w

2

w – 25

3

3w + 2

4

w + 22

5

2w

6

2w + 18

Express the number of students in Grades 1 to 3 in terms of w.

Number of students in Grades 1 to 3 = w + w – 25 + 3w + 2 = 5w – 25 +2 = 5w – 23 Express the number of students in Grades 4 to 6 in terms of w.

Number of students in Grades 4 to 6 = w + 22 + 2w + 2w + 18 = 5w + 22 + 18 = 5w + 40 Find the total number of students enrolled in Grades 1 to 6 at Bright Sparks STEM School when w = 85.

When w = 85, Total students enrolled = = = = = = =

students in Grades 1 to 3 + students in Grades 4 to 6 5w – 23 + 5w + 40 10w – 23 + 40 10w + 17 10 x 85 + 17 850 + 17 867

When w = 85, the total number of students enrolled in Grades 1 to 6 at Bright Sparks STEM School is 867.

85

Let’s Practice 1. Keira has $x. Riley has 3 times as much money as Keira. Riley's mother gives Riley an additional $20. (a) Express the amount of money Riley has in terms of x. (b) How much money does Riley have when x = 50?

2. Blake, Wyatt and Sophie each have a pet cat. Blake's pet cat weighs

2z kg. Wyatt's cat is twice as heavy as Blake's cat. Sophie's cat is half the weight of Blake's cat. (a) Express the total weight of the cats in terms of z.

1 (b) Find the weight of each cat when z = 2 .

2

86

3. The width of a rectangle is r cm.

The length of the rectangle is 3 times its width. (a) Express the perimeter of the rectangle in terms of r. (b) Find the perimeter of the rectangle when r = 8. (c) Draw the rectangle and label the length and width when r = 3.

87

4. The side lengths of a triangle are t cm, (t + 5) cm and 2t cm. (a) Express the perimeter of the triangle in terms of t. (b) Find the perimeter of the triangle when t = 15. (c) Draw the triangle and label the length of each side when t = 5.

88

5. Halle, Sophie and Chelsea put their savings together. Halle contributed $r. Sophie contributed 5 times more than Halle. Chelsea contributed $30 less than Sophie. (a) Express the amount each child contributed in terms of r. (b) Find the amount each child contributed when r = 15. (c) Find the total amount of money when r = 126.

89

6. Blake bought d cans of cat food for $6 each. He paid for the cat food with a $100 note. (a) Express the change Blake will receive in terms of d. (b) What change will Blake receive when d = 9? (c) Blake decides to also purchase a cat collar for $20.

Express the change Blake will receive in terms of d.

90

Solve It! The perimeter of a rectangle is 56 cm.

The length of the rectangle is p cm. (a) (b) (c) (d)

Express the width of the rectangle in terms of p.

Express the area of the rectangle in terms of p.

Find the area of the rectangle when p = 18.

What shape is formed when p = 14?

91

At Home 1. Riley has 5 kg of apples. She goes to the supermarket and buys 5 bags of apples, each with a mass of u kg. (a) Express the mass of the apples Riley has in terms of u. (b) What is the total mass of the apples Riley has if u = 2?

2. Chelsea wanted to buy f movie tickets for her friends. The tickets cost $13 each. The cashier told Chelsea she needed another $6 to buy the movie tickets. (a) Express the amount of money Chelsea had in terms of f. (b) How much money did Chelsea have if f = 5?

92

3. Wyatt, Jordan and Ethan each have a length of rope. Wyatt's rope is 8p cm length. Jordan's rope is half as long as Wyatt's. Ethan's rope is 40 cm longer than Jordan's rope. (a) Express the length of each child's rope in terms of p. (b) Find the length of each child's rope if p = 22. (c) Find the total length of all the ropes if p = 17 .

93

Looking Back 1. Mr. Whyte buys a used car for $x. He sells the used car to his brother for $2,350 less than he paid for it. Express the price Mr. Whyte's brother paid for the car in terms of x.

2. Ethan scored m points on his mathematics quiz. Blake scored 23 points more than Ethan. What was Blake's score in terms of m?

3. Evaluate the algebraic expressions when a = 3 and b = 15. (a) 75 – b + 5a

(b) 8b – 8a + 33 – 11

94

4. Complete the table.

Expression

Value of Expression when d = 12

Value of Expression when d = 5

2d +8 12d – 24 – 3 10 + 4d 2 331 – 12d 12d + 8 4 5. Simplify. Show your working. (a) 33 + 2x + 4x – x

(b) 22 + 9p + 13 – 2p – p

95

6. Riley and Chelsea are making pancakes for dessert.

Chelsea made w pancakes. Riley made 16 more pancakes than Chelsea.

They made 56 pancakes in all. (a) Express the number of pancakes Riley made in terms of w. (b) How many pancakes did Riley make?

7. A bricklayer has 25 kg of cement. He buys 12 bags of cement.

Each bag has a mass of h kg. (a) Express the total mass of the cement in terms of h. (b) What is the total mass of the cement if h = 32? (c) What is the total mass of the cement if h = 11?

96

8. Ethan has $44. He buys q donuts for his friends for $3 each and a cupcake for his sister for $2. (a) Express the amount of money Ethan has left in terms of q. (b) How much money does Ethan have left if q = 5? (c) How much money does Ethan have left if q = 13?

97

3

Fractions

Anchor Task

Cheesecake Recipe Serves 10 people Prep Time 30 minutes Cook Time 1 hour Cooling Time 15 minutes

Ingredients Cheesecake 1 2 lbs cream cheese 2 3 cup sour cream 4 1 tsp salt 2 3 1 cups sugar 4 1 3 tbsp flour 2 5 eggs 2 egg yolks

98

Crust 14 Graham Crackers 1 1 cup pecans 2 4 tbsp butter 1 cup sugar 4 1 tsp cinnamon 2

Multiplying Fractions Let’s Learn Riley is making fruit punch for the school fair.

The recipe requires a

4 cup of lemon juice per liter.

5

She plans on making 4 liters of fruit punch.

How much lemon juice will she need in total?

4 5 4x 4 5 Multiply the numerator by the whole number.

Then simplify.

31 5 4 x 4 = 4 x 4 5 5 = 16 = 3 1 5 5

When multiplying a fraction by a whole number, we multiply the numerator by the whole number. Then simplify if possible.

Riley needs 3

1 cups of lemon juice in total.

5

99

Multiply

7 by 6.

8 Remember to write the fraction in its simplest form.

7 6x7 6x = 8 8 42 = 8 2 1 = 5 = 5 8 4

When multiplying a proper fraction by a whole number, the product is less than the whole number.

A bowling ball has a mass of 4

5 kg. Find the mass of 3 such bowling balls.

7 ?

4

5 7

12 5 33 3 x 4 = 3 x 7 7 3 x 33 = 7 99 = 7 1 = 14 7 3 bowling balls have a total mass of 14

15 = 2 1 7 7 5 4 7 is a mixed number. So we expect the product to be greater than 3!

1 kg.

7

When multiplying a mixed number by a whole number, we convert the mixed number into an improper fraction. Then we multiply and simplify.

100

3 Find 7 x 5 .

8 3 43 =7x 8 8 7 x 43 = 8 301 = 8 5 = 37 8 7x5

7x5

3 8 3 0 2 4 6 5

7 1

3 40 + 3 43 58 = 8 = 8

1 6 5

3 5 = 37 8 8

3 cm. A total of 18 such textbooks are 4 stacked in a pile in a bookstore. Find the height of the stack of books.

A textbook has a width of 3

3 by 18 to find the total height of the books.

4 270 ÷ 4 is 67 R 2.

3 12 + 3 1 3 x 18 = x 18 The product is 67 2 .

4 4 15 1 5 6 7 = x 18 x 1 8 4 4 2 7 0 270 1 2 0 2 4 = 1 5 0 3 0 4 2 7 0 2 8 2 1 = 67 = 67 2 4 2 1 The stack of books is 67 cm high.

2 Let's multiply 3

1 01

Blake bought

3 of an apple pie to school to share with his friends. They ate 5

2 of the pie Blake brought. What fraction of the whole apple pie did Blake 3 and his friends eat?

3 of whole 5

When both factors are proper fractions, the product is less than both factors.

2 3 of 3 5

3 2 3x2 x = 5 3 5x3 6 2 = = 15 5

When multiplying a fraction by a fraction, multiply the numerators and the denominators. Then simplify if possible.

Blake and his friends ate

1 02

2 of the whole apple pie.

5

Find

1 3 of .

4 4

1 3 1x3 x = 4 4 4x4 3 = 16

3 4

1 3 3 of is .

4 4 16

1 3 of 4 4

Find the product of 4 3 4x3 x = 5 8 5x8 12 = 40 3 = 10

The product of

4 3 and .

5 8

3 8

4 3 3 and is .

5 8 10

Find the product of

4 3 of 5 8

3 8 and .

4 5

3 8 3x8 x = 4 5 4x5 24 = 20 1 = 1 5 The product of

8 5 is an improper fraction.

3 8 1 and is 1 .

4 5 5

103

Let’s Practice 1. Complete the following. Show your working and write your answer in its simplest form. (a)

4 5 x 3 (b) 4 x 5 8

(c)

2 x 6 (d) 3

(e) 12 x

104

3 x8 7

3 7

(f) x 13 4 12

2. Multiply the fractions. Show your working and write your answer in its simplest form. (a) 8 x

2 (b) 3

(c)

3x

8 3 (d) 6 x 9 4

(e)

7 3 x 5 (f) 8 x 12 5

(g) 4 x

7 (h) 20

6 x4 7

4 x 12 15

105

3. Multiply the mixed numbers. Show your working and write your answer in its simplest form. (a) 3

1 x4 2

(b) 5 x 3

(c)

106

2

3 4

7 x3 8

4. Multiply the mixed numbers. Show your working and write your answer in its simplest form. (a) 5

(c)

2 x 2 7

4x4

(e) 5

5

6

3 x 15 5

2 (g) 20 x 4 3

(b) 2

2 x6 3

(d) 5 x 12

(f) 6

1 2

7 x4 10

(h) 12 x 3

4 5

107

5. Color parts of the rectangle to show the product of the fractions.

Write the product in its simplest form. (a)

1 2 x 2 3

(b)

1 3 x 3 4

(c)

2 1 x 3 3

(d)

4 2 x 5 5

(e)

7 2 x 8 3

(f)

4 5 x 5 6

108

6. Multiply the fractions. Show your working and write your answer in its simplest form. (a)

1 2 x (b) 2 3

4 1 x 5 2

(c)

5 3 x (d) 9 4

5 4 x 3 5

(e)

9 2 x (f) 2 3

7 2 x 9 3

(g)

5 4 x (h) 3 11

5 8 x 6 3

109

Hands On Work in pairs. (a) Use the grid below to draw a rectangle. Lightly shade rectangle blue. Color

1 of the 2

1 of the shaded part green. Write the fraction of 3

the rectangle that is colored green.

(b) Use the grid below to draw a rectangle. Lightly shade rectangle yellow. Color

3 of the shaded part red. Write the fraction of 4

the rectangle that is colored red.

110

1 of the 4

(c) Use the grid below to draw a rectangle. Lightly shade rectangle yellow. Color

5 of the 6

2 of the shaded part blue. Write the fraction of 3

the rectangle that is colored blue.

(d) Use the grid below to draw a rectangle. Lightly shade rectangle red. Color

5 of the 8

3 of the shaded part green. Write the fraction of 4

the rectangle that is colored green.

111

At Home 1. Complete the following. Show your working and write your answer in its simplest form. (a)

4 2 x 5 (b) 6 x 7 3

(c)

5 5 x 12 (d) 10 x 8 6

(e) 6 x

112

3 (f) 7

7 x5 9

2. Multiply the mixed numbers. Show your working and write your answer in its simplest form. (a) 3

2 x5 5

(b) 2 x 3

5 8

2 4 x 2 3

(d) 7 x 3

3 5

1 (e) 12 x 4 8

(f) 8 x 5

3 12

(c)

113

3. Color parts of the rectangle to show the product of the fractions.

Write the product in its simplest form. (a)

3 2 x 4 3

(b)

2 1 x 7 4

(c)

4 3 x (d) 7 4

5 3 x 8 4

4. Multiply the fractions. Show your working and write your answer in its simplest form. (a)

114

9 5 x (b) 4 6

17 3 x 4 7

Solve It! 1 as many figurines as 4 1 Sophie. Sophie gives Chelsea 20 of her figurines. Chelsea now has as many 2 Sophie and Chelsea collect figurines. Chelsea has

figurines as Sophie. How many figurines did Sophie have originally?

115

Fractions and Division Let’s Learn Blake pours 2 liters of mineral water into 5 cups. Each cup has the same volume of water. Find the volume of mineral water in each cup.

2 ÷ 5 = 1 of 2 5

= 2 5

Each cup contains

116

2 5 of 1,000 ml 2,000 = 5 = 400 ml

2 liters of mineral water.

5

A recipe requires 16 teaspoons of cocoa powder to make 12 cookies.

How many teaspoons of cocoa powder are there in each cookie?

1 of 16 12 16 = 12 4 1 = = 1 3 3 16 ÷ 12 =

Each cookie contains 1

1 teaspoons of cocoa powder.

3

Sophie has a strip of paper

3 m in length.

4

She cuts the strip into 6 pieces of equal length to make bookmarks.

Find the length of each bookmark.

3 m 4

1 3 of m 6 4 3 1 3 ÷ 6 = of 4 6 4 1 3 = x 6 4 3 1 = = 24 8

Each bookmark has a length of

Dividing by 6 is the same as 1 multiplying by 6 !

1 m.

8

117

Find

3 ÷ 5.

4

3 1 3 ÷ 5 = of 4 5 4 1 3 = x 5 4 3 = 20

3 4 1 3 of 5 4

3 3 ÷5= 4 20

Keira is baking chocolate chip cookies.

Each cookie requires

2 cup of chocolate chips.

7

Keira has 3 cups of chocolate chips.

How many whole cookies can she make?

2 7

2 7

2 7

1 cup

2 7

2 7

2 7

1 cup

2 7 =3x 7 2 7x3 = 2 21 = 2 1 = 10 2

2 7

2 7

2 7

2 7

1 cup

3÷

Keira can make 10 whole cookies.

118

2 Dividing by 7 is the same 7 as multiplying by 2 !

A phone charger requires

5 m of cable. How many chargers can be made 8

with 10 m of cable?

5 8 = 10 x 8 5 80 = 5 10 ÷

= 16 16 phone chargers can be made with 10 m of cable.

Find 8 divided by

3 .

5

3 5 =8x 5 3 40 = 3 1 = 13 3 8÷

8 divided by

3 1 = 13 5 3

119

Let’s Practice 1. Complete the following.

Show your working and write your answer in its simplest form. (a) 9 bags of flour are used to make 36 cakes. What fraction of a bag of flour is used in 1 cake?

(b) 16 mini pizzas are ordered to feed 6 guests at a party. Each guest received an equal amount of pizza. How much pizza does each guest receive?

(c)

12 ÷ 8 (d) 10 ÷ 4

(e) 8 ÷ 20 (f) 9 ÷ 27

120

2. Use the model to help divide the fractions.

Write the answer in its simplest form. (a)

2 ÷ 5 3

(b)

7 ÷4 8

3. Complete the following. Show your working and write your answer in its simplest form. (a)

1 ÷ 12 (b) 2

3 ÷6 8

(c)

8 ÷ 4 (d) 3

11 ÷9 5

(e)

4 ÷ 8 (f) 7

2 ÷ 12 3

121

4. Use the model to help divide whole numbers by fractions.

Write the answer in its simplest form. (a) 4 ÷

2 5

(b) 4 ÷

2 9

5. Complete the following.

Show your working and write your answer in its simplest form. (a) 10 ÷

1 3 (b) 8 ÷ 2 4

(c)

2 2 (d) 9 ÷ 5 7

1 22

12 ÷

Solve It! 3 of a packet of nuts. He divides the nuts equally into 12 bags.

8 He gives 6 bags to his family and 2 bags to his friends. He keeps the rest for himself. What fraction of the original packet does Ethan keep for himself?

Ethan has

123

At Home omplete the following. Show your working and write your answer in C its simplest form. (a) 3 ÷ 12 (b) 10 ÷ 6

(c)

4 ÷ 20 (d) 6 ÷ 16

(e) 12 ÷ 8 (f) 4 ÷ 18

(g)

2 ÷ 6 (h) 3

(i)

7 8 ÷ 10 (j) ÷4 2 3

124

6 ÷8 7

(k)

5 3 ÷ 4 (l) ÷ 12 4 10

(m)

14 ÷ 7 (n) 3

4 ÷ 12 9

(o) 2 ÷

4 2 (p) 8 ÷ 7 3

(q) 12 ÷

5 3 (r) 7 ÷ 3 4

(s) 9 ÷

7 18 (t) 10 ÷ 3 7

125

Word Problems Let’s Learn Riley has

2 liters of milk. She uses 7

3 of the milk to make a milkshake.

4 How much milk did she use?

3 2 of 4 7

3 2 x 4 7 6 = 28 3 = 14 =

Riley used

2 l 7

3 l of milk.

14

3 2 of l 4 7

24 mini pizzas are divided equally among 18 people. How many mini pizzas does each person receive?

24 18 4

= 3 1

=1 3 24 ÷ 18 =

Each person receives 1

126

1 mini pizzas.

3

A plank of wood has a length of

3 m. It is cut into 6 pieces 4

of equal length. Find the length of each piece of wood.

We need to divide

3 m by 6.

4

3 1 3 ÷ 6 = of 4 6 4 1 3 = x 6 4 1 3 = = 24 8 Each piece of wood has a length of

A can of beans has a mass of 2

1 m.

8

4 kg.

5

Find the mass of 12 such cans.

4 14 12 x 2 = 12 x 5 5 12 x 14 = 5 168 = 5 3 = 33 5

1 x 1 4 1 2 1 6

2 4 8 0 8

3 5 1 6 1 5 1 1

12 cans of beans has a mass of 33

3 8 8 5 3

3 kg.

5

127

2 of his money on 5 1 some new ear pods. He then spent of his 8 Ethan had $360. He spent

remaining money on a phone case. Find the cost of the ear pods and the phone case. $360 spent on ear pods

1 3 He spent 8 of 5 of his money on a phone case.

? spent on phone case ?

Let's find the cost of the ear pods.

2 2 x 360 of 360 = 5 5 720

= 5

= 144

1 5 7 5 2 2

The ear pods cost $144.

4 4 2 0 2 0 2 0 2 0 0

Now let's find the cost of the phone case.

Method 1 Find

1 of the remaining money.

8

$360 – $144 = $216 1 1 x 216 of 216 = 8 8 = 27 The phone case costs $27.

1 28

2 8 2 1 1 6 5 5

7 6 6 6 0

Method 2 Find

1 3 of the money Ethan had originally.

8 5

1 3 1 3 of = x 8 5 8 5 3

= 40 3 3 x 360 of 360 = 40 40 1080 108 = = 40 4 = 27 The ear pods cost $144 and the phone case costs $27.

Sophie is pouring water from a cooler into cups. The cooler contains 18 liters of water and each cup can hold

3 liters of water. How many cups can she 7

fill with the 18 liters of water?

We need to divide 18 by

3 .

7

3 7 = 18 x 7 3 126

= 3 18 ÷

= 42 Sophie can fill 42 cups of water.

129

Let’s Practice 3 of them are girls.

4 (a) How many boys are at the museum? (b) How many more girls than boys are there?

1. 56 children are at the museum.

1 2. Keira made 168 cookies to sell at the fair. She sold of them on 3 5 Saturday and of the remaining cookies on Sunday.

8 How many cookies did she sell on Sunday?

130

3 min to complete a full rotation.

4 How long does it take to complete 12 rotations?

3. It takes a Ferris wheel 8

4. Mr. Timmins is tiling his bathroom floor which has a length of 4

3 m and 8

a width of 4 m. (a) Find the area of Mr. Timmins' bathroom. (b) The tiles cost $28 per square meter. How much will it cost to tile the bathroom?

131

5. Michelle spent

3 1 of her savings on a telescope. She spent of the 5 4

remaining money on a chess set. (a) What fraction of her total savings did she spend on the chess set? (b) She had $102 left over. How much did the telescope cost?

1 6. Halle picked some flowers in her garden. of the flowers were roses, 3 1 of them were tulips and the rest were daisies. If she picked 20 4 daisies, how many flowers did she pick in all?

1 32

At Home 3 1. Ethan has 140 toy cars. He gives his brother of the cars. How many 7 cars does he have left?

4 2. Sophie is making a poster for a school presentation. She colors of the 9 3 poster blue and of the remaining part green. What fraction of the 4 poster is green?

133

3. Mrs. Laycock's lawn is rectangular in shape with a length of 12 m and a breadth 7

5 m.

6

(a) Find the area of Mrs. Laycock's lawn. (b) She plans on replacing the lawn with synthetic grass.

The synthetic grass costs $17 per square meter.

How much will it cost to replace the lawn?

2 min to complete a full rotation.

3 How long does it take to complete 36 rotations?

4. It takes a merry-go-round 1

1 34

Looking Back 1. Multiply the fractions and mixed numbers.

Show your working and write your answer in its simplest form. (a)

4 7 x 6 (b) 2 x 5 9 8

(c)

8x3

(e)

7 5 7 2 x (f) x 2 6 12 3

2

7

(d) 12 x 4

3 4

135

2. Complete the following.

Show your working and write your answer in its simplest form. (a) 8 ÷ 12 (b) 24 ÷ 18

(c)

3÷

3 (d) 10 ÷ 36 4

(e) 14 ÷ 16 (f) 12 ÷

2 3

(g)

3 ÷ 4 (h) 5

(i)

8 9 ÷ 10 (j) ÷7 3 4

136

6 ÷3 7

3. 39 liters of olive oil is to be poured into smaller bottles. Each bottle can hold

3 liters of oil. How many bottles can be filled with 39 liters?

4

2 of the apples were red.

7 (a) How many green apples were there? (b) How many more green apples than red apples were there?

4. Sophie bought 182 red and green apples.

137

4

Ratio

Ratio and Fraction Anchor Task

138

Let’s Learn In Blake's fruit bowl there are 2 oranges and 5 pears.

Let's use a model to represent the number of each type of fruit. oranges pears

The ratio of the number of oranges to the number of pears is 2 : 5.

The ratio of the number of pears to the number of oranges is 5 : 2.

We can also express ratio as a fraction.

Number of oranges 2 = 5 Number of pears The number of oranges is

2 the number of pears.

5

5 Number of pears = Number of oranges 2 The number of pears is

5 the number of oranges.

2

There are 7 fruits in the bowl in total. The ratio of oranges to the total number of fruits is 2 : 7. So,

2 of the fruits in the bowl are oranges.

7

The ratio of pears to the total number of fruits is 5 : 7. The number of pears is

5 of the total number of fruits in the bowl.

7

139

Riley and Keira put their money together to buy a $50-giftcard for Sophie.

Riley contributed $30 and Keira contributed $20.

1 unit

Each unit has a value of $10.

Riley $50 Keira

The ratio of Riley's contribution to Keira's contribution is 3 : 2.

The ratio of Keira's contribution to Riley's contribution is 2 : 3.

Each unit has a value of $10. The total cost of the giftcard is $50 = 5 units.

The ratio of Riley's contribution to the total cost of the giftcard is 3 : 5.

Riley paid for

3 of the giftcard.

5

The ratio of Keira's contribution to the total cost of the giftcard is 2 : 5.

Keira paid for

2 of the giftcard.

5

The model below represents the lengths of 2 pencils.

1 cm Yellow pencil Green pencil

The yellow pencil is 6 cm in length. The green pencil is 8 cm in length.

The ratio of the length of the yellow pencil to the length of the green pencil is 6 : 8.

We express the ratio 6 : 8 in its simplest form by dividing each term in the ratio by the greatest common factor. ÷2

6 : 8

÷2

= 3 : 4 The ratio 6 : 8 in its simplest form is 3 : 4.

140

In its simplest form, the ratio of the length of the green pencil to the length of the yellow pencil is 4 : 3.

The green pencil is 4 the length of the 3 yellow pencil.

The yellow pencil is 3 the length of the 4 green pencil.

The total length of both pencils is 14 cm. Let's find the ratio of the length of each pencil to the total length of the pencils.

The length of the yellow pencil to the total length of the pencils is 6 : 14.

The length of the green pencil to the total length of the pencils is 8 : 14. ÷2

6 : 14

÷2

= 3 : 7

÷2

8 : 14

÷2

= 4 : 7

The length of the yellow pencil to the total length of the pencils is 3 : 7.

The yellow pencil is

3 of the total length of the pencils.

7

The length of the green pencil to the total length of the pencils is 4 : 7.

The green pencil is

4 of the total length of the pencils.

7

1 41

The mass of Ethan and his siblings is shown in the model.

1 unit Mike Ethan Jessica

The ratio of the mass of Jessica to the mass of Ethan is 3 : 7.

Jessica's mass is

3 of Ethan's mass.

7

The ratio of the mass of Mike to the mass of Ethan is 11 : 7.

Mike's mass is

11 of Ethan's mass.

7

The ratio of the mass of Jessica to the mass of Mike 3 : 11.

Jessica's mass is

3 of Mike's mass.

11

Ethan and his siblings have a total mass of 21 units.

Jessica's mass is ÷3

3 : 21

3 of the total mass of the siblings.

21

÷3

= 1 : 7 1 Jessica's mass is of the total mass of the siblings.

7 Ethan's mass is ÷7

7 : 21

÷7

= 1 : 3 Ethan's mass is

142

7 of the total mass of the siblings.

21

1 of the total mass of the siblings.

7

The ages of Riley, Tejal and Zhang Wei is in the ratio 2 : 3 : 6.

Riley Tejal Zhang Wei

The ratio of the age of Riley to the age of Zhang Wei is 2 : 6 = 1 : 3.

The age of Riley is

1 the age of Zhang Wei. 3

The age of Zhang Wei is

3 the age of Riley.

1

We can say Zhang Wei is 3 times older than Riley.

The ratio of the age of Tejal to the age of Zhang Wei is 3 : 6 = 1 : 2.

The age of Tejal is

1 the age of Zhang Wei. 2

The age of Zhang Wei is

2 the age of Tejal.

1

Zhang Wei is twice as old as Tejal.

The ratio of the age of Riley to the combined age of Tejal and Zhang Wei is 2 : 9.

The age of Riley is

2 the combined age of Tejal and Zhang Wei. 9

The ratio of the age of Zhang Wei to the combined age of all three children is 6 : 11.

The age of Zhang Wei is

6 the combined age of all three children.

11

143

Let’s Practice 1. Complete the tables. (a) Express each ratio as a fraction.

Ratio

Fraction

1:4 7:9 9:5 2:5 12 : 5

(b) Express each fraction as a ratio.

Fraction 7 1 1 3 5 7 13 9 4 5

144

Ratio

2. Express each of the following quantity comparisons as a ratio in its simplest form. Show your working. (a) $70 to $90

(b) 12 cm to 50 cm

(c)

27 kg to 9 kg

(d) 35 ml to 140 ml

(e) 15 min to 1 h

(f) 14 in to 63 in

145

3. An eraser has a length of 3 cm. A stapler has a length of 7 cm. eraser stapler

(a) The ratio of the length of the eraser to the length of the stapler is

:

.

(b)

Length of eraser = Length of stapler

(c)

The length of the eraser is

the length of the stapler.

(d) The ratio of the length of the stapler to the length of the eraser is (e)

:

.

Length of stapler = Length of eraser

(f) The length of the stapler is

the length of the eraser.

(g) The ratio of the length of the eraser to the total length of both objects is

:

.

(h) The length of the eraser is

the total length of both objects.

(i) The ratio of the length of the stapler to the total length of both objects is

:

(j) The length of the stapler is

146

. the total length of both objects.

4. Farmer Joe keeps cows and horses in the ratio 9 : 11. cows horses

(a) The ratio of the number of horses to cows is (b)

Number of horses = Number of cows

(c)

The number of horses is

.

the number of cows.

(d) The ratio of the number of cows to horses is (e)

:

:

.

Number of cows = Number of horses

(f) The number of cows is

the number of horses.

(g) The ratio of the number of cows to the total number of animals is

:

.

(h) The number of cows is

the total number of animals.

(i) The ratio of the number of horses to the total number of animals is

:

.

(j) The number of horses is

the total number of animals.

147

5. The mass of a goose is 3 kg. The mass of a lamb is 9 kg. goose lamb

Express all ratios in simplest form. (a) Find the ratio of the mass of the lamb to the mass of the goose.

The mass of the lamb is

the mass of the goose.

(b) Find the ratio of the mass of the goose to the mass of the lamb.

The mass of the goose is

the mass of the lamb.

(c) Find the ratio of the mass of the goose to the total mass of both animals.

The mass of the goose is

the total mass of both animals.

(d) Find the ratio of the mass of the lamb to the total mass of both animals.

The mass of the lamb is

1 48

the total mass of both animals.

6. The model shows the number of flowers in Chelsea's garden.

1 unit roses tulips daisies

Express all ratios in simplest form. (a) Find the ratio of the number of roses to daisies.

The number of roses is

the number of daisies.

(b) Find the ratio of the number of tulips to roses.

The number of tulips is

the number of roses.

(c) Find the ratio of the number of roses and daisies to the number of tulips.

The number of tulips is

the number of roses and daisies.

(d) Find the ratio of the number of daisies to the total number of flowers in Chelsea's garden.

The number of daisies is Chelsea's garden.

the total number of flowers in

149

7. The capacity of 3 beakers is in the ratio 8 : 2 : 4 Beaker A Beaker B Beaker C

Express all ratios in simplest form. (a) Find the ratio of the capacity of Beaker B to the capacity of Beaker C.

The capacity of Beaker B is

the capacity of Beaker C.

(b) The capacity of Beaker C is

the capacity of Beaker B.

(c) The capacity of Beaker C is Beaker B.

times the capacity of

(d) Find the ratio of the capacity of Beaker B to the capacity of Beaker A.

The capacity of Beaker B is

the capacity of Beaker A.

(e) The capacity of Beaker A is

the capacity of Beaker B.

(f) The capacity of Beaker A is Beaker B.

times the capacity of

150

8. Sophie has

1 the savings of Halle.

4

Sophie Halle

(a) Express Halle's savings to Sophie's savings as a ratio.

(b) Express Halle's savings as a fraction of Sophie's savings.

(c)

Halle has

9. Jordan ran

times more savings that Sophie.

7 the distance Ethan ran.

1

Jordan Ethan

(a) Express the distance Jordan ran to the distance Ethan ran as a ratio.

(b) Express the distance Ethan ran as a fraction of the distance Jordan ran.

(c)

Jordan ran

times further than Ethan.

1 51

10. Riley read 15 books. Blake read 5 books.

(a) Find the ratio of the number of books Riley read to the number of books Blake read. Write the ratio in its simplest form.

(b) Express the number of books Blake read as a fraction of the number of books Riley read. Write the fraction in its simplest form.

(c) Express the number of books Riley read as a fraction of the number of books Blake read. Write the fraction in its simplest form.

(d) Riley read

times the number of books Blake read.

(e) Express the number of books Blake read as a fraction of the total number of books read by Riley and Blake. Write the fraction in its simplest form.

152

11. Ethan has 5 times more money than his brother, Steve.

(a) What is the ratio of Ethan's money to Steve's money?

(b) What fraction of Ethan's money does Steve have?

(c) Ethan and Steve put their money together.

What fraction of the money came from Ethan?

(d) What fraction of the money came from Steve?

1 53

12. Chelsea's height is

4 Sophie's height.

5

(a) What is the ratio of Sophie's height to Chelsea's height?

(b) Express Sophie's height as a fraction of Chelsea's height.

(c) Express Chelsea's height as a ratio of Chelsea and Sophie's combined height.

(d) Express Sophie's height as a fraction of Sophie's and Chelsea's combined height.

154

Hands On 1. Place 20 blue cubes into a cup. Place 20 red cubes into another cup Take turns to close your eyes and remove a small handful of cubes from each cup.

(a) Express the number of blue cubes to the number of red cubes as a ratio in its simplest form. (b) Express the number of red cubes as a fraction of the number of blue cubes.

Repeat 5 times.

2. Use blue and red cubes to show each ratio in the table.

Remove the cubes to show the ratio in its simplest form.

Blue Cubes

Red Cubes

3

9

6

18

2

12

5

20

4

20

Ratio

Ratio in Simplest Form

155

At Home 1. Complete the tables. (a) Express each ratio as a fraction.

Ratio

Fraction

2:7 13 : 9 7:9 5:1 12 : 7

(b) Express each fraction as a ratio.

Fraction 9 1 1 5 5 11 13 7 5 3

156

Ratio

2. Express each of the following quantity comparisons as a ratio in its simplest form. Show your working. (a) 6 cm to 48 cm

(b) 20 min to 2 h

(c)

36 lb to 6 lb

(d) 90 l to 10 l

(e) 500 g to 2 kg

(f) 10 in to 2 ft

1 57

3. Keira is 11 years old. Her younger sister, Shanice, is 7 years old.

Shanice Keira

(a) The ratio of Shanice's age to Keira's age is

:

.

:

.

(b) Shanice's age = Keira's age (c)

Shanice is

the age of Keira.

(d) The ratio of Keira's age to Shanice's age is (e)

Keira's age = Shanice's age

(f) Keira is

the age of Shanice.

(g) The ratio of Shanice's age to the combined age of her and her sister is

:

(h) Shanice is her sister.

. the age of the combined ages of her and

(i) The ratio of Keira's age to the combined age of her and her sister is

:

(j) Keira is her sister.

the age of the combined ages of her and

158

.

4. The mass of a bag of apples is 2 kg. The mass of a bag of potatoes is 10 kg. apples potatoes

Express all ratios in simplest form. (a) Find the ratio of the mass of the apples to the mass of the potatoes.

The mass of the apples is

the mass of the potatoes.

(b) Find the ratio of the mass of the potatoes to the mass of the apples.

The mass of the potatoes is

the mass of the apples.

(c) Find the ratio of the mass of the apples to the total mass of apples and potatoes.

The mass of the apples is and potatoes.

the total mass of apples

(d) Find the ratio of the mass of the potatoes to the total mass of apples and potatoes.

The mass of the potatoes is and potatoes.

the total mass of apples

159

5. The model shows the fish Wyatt caught on a boat trip.

1 unit cod mullet bream

Express all ratios in simplest form. (a) Find the ratio of the number of cod to the number of bream.

The number of cod Wyatt caught is bream he caught.

the number of

(b) Find the ratio of the number of mullet to the number of bream.

The number of mullet Wyatt caught is bream he caught.

the number of

(c) Find the ratio of the number of cod and bream to the number of mullet.

The number of cod and bream Wyatt caught is number of mullet Wyatt caught.

160

the

6. To make a fruit punch, Halle mixed orange juice, apple juice and pineapple juice in the ratio 12 : 4 : 3 Orange Apple Pineapple

Express all ratios in simplest form. (a) Find the ratio of the volume of orange juice to the volume of apple juice.

The volume of orange juice is

the volume of apple juice.

(b) The volume of apple juice is

the volume of orange juice.

(c) In Halle's fruit punch there is as apple juice.

times as much orange juice

(d) Find the ratio of the volume of pineapple juice to the volume of orange juice.

The volume of pineapple juice is orange juice. (e) The volume of orange juice is pineapple juice. (f) In Halle's fruit punch there is as pineapple juice.

the volume of the volume of times as much orange juice

161

7. The mass of a hamster is

1 the mass of a kitten.

3

hamster kitten

(a) What is the ratio of the mass of the kitten to the mass of the hamster?

(b) Express the mass of the kitten as fraction of the mass of the hamster.

The kitten is 8. Riley's house is

times heavier than the hamster.

6 the distance from school compared to Michelle's house.

1

Riley Michelle

(a) Express the distance of Michelle's house from school to the distance of Riley's house from school as a ratio.

(b) Express the distance of Michelle's house from school as a fraction of the distance of Riley's house from school.

Michelle's house is

1 62

times closer to school than Riley's house.

9. Jordan picked 18 apples. Blake picked 6 apples.

(a) Find the ratio of the number of apples Jordan picked to the number of apples Blake picked. Write the ratio in its simplest form.

(b) Express the number of apples Blake picked as a fraction of the number of apples Jordan picked.

Write the fraction in its simplest form.

(c) Express the number of apples Jordan picked as a fraction of the number of apples Blake picked.

Write the fraction in its simplest form.

Jordan picked

times as many apples than Blake.

(d) Express the number of apples Blake picked as a fraction of the total number of apples picked by Jordan and Blake.

Write the fraction in its simplest form.

163

10. Chelsea has 10 times more marbles than Halle.

(a) What is the ratio of the number of Chelsea's marbles to the number of Halle's marbles?

(b) Express the number of marbles Halle has as a fraction of the number of marbles Chelsea has.

(c) Chelsea and Halle put all of their marbles in a jar.

What fraction of the marbles in the jar are Halle's?

(d) What fraction of the marbles in the jar are Chelsea's?

164

11. Ethan has

7 the money Wyatt has.

13

(a) What is the ratio of Wyatt's money to Ethan's money?

(b) Express Wyatt's money as a fraction of Ethan's money.

(c) Ethan and Wyatt put their money together to buy a sandwich.

What fraction of the money is Ethan's?

What fraction of the money is Wyatt's?

165

Solve It! 1. Riley has

5 the amount of savings Chelsea has.

6

(a) Draw a model to compare the amount of money each child has.

(b) Express Chelsea's savings as a fraction of Riley's savings.

(c)

Chelsea gives

1 of her savings to Riley.

3

What is the ratio of Chelsea's savings to RiIey's now?

166

2. Ethan has

3 the number of computer games Wyatt has.

11

(a) Draw a model to compare the number of video games each child has.

(b) Express the number of video games Wyatt has as a fraction of the number of video games Ethan has.

3 of his video games to Ethan. Write the ratio of the 11 number of video games Ethan has to the number of video games Wyatt has now. Express the ratio in its simplest form.

(c) Wyatt gives

1 67

Ratio and Proportion Let’s Learn Sophie is making orange soda. To do so, she mixes orange juice with soda water at the ratio 1 : 2. To make a small jug of orange soda, she mixes 1 cup of orange juice with 2 cups of soda water. orange juice soda water

Sophie would like to make orange soda for all of her friends. To do so, she'll need to mix orange juice and soda water in the same ratio, but in different amounts. She uses the table below to help her.

Cups of Orange Juice

1

2

4

8

Cups of Soda

2

4

8

16

If Sophie uses 2 cups of orange juice, she'll need 4 cups of soda water.

If she uses 16 cups of soda water, she'll need 8 cups of orange juice. To make different amounts of orange soda, the amount of orange juice and soda water she uses changes, but the ratio and proportion are the same.

To paint a tennis court, a painter needs to mix yellow paint and blue paint in the ratio 1 : 4. yellow paint blue paint

Tins of Yellow Paint

1

5

10

20

Tins of Blue Paint

4

20

40

80

If the painter uses 5 tins of yellow paint, he'll need 20 tins of blue paint.

If he uses 80 tins of blue paint, he'll need 20 tins of yellow paint.

168

To make cupcakes, a baker needs to mix flour and sugar. The table shows the quantity of each ingredient needed to make different quantities of cupcakes.

Number of Cupcakes 50

100

150

200

250

Flour (kg)

3

6

9

12

15

Sugar (kg)

1

2

3

4

5

3:1

3:1

3:1

3:1

3:1

3 1

3 1

3 1

3 1

3 1

Flour : Sugar Fraction of Flour to Sugar

The ratio of flour to sugar is 3 : 1. For each amount of flour the baker uses, 1 he'll need as much sugar.

3 If the baker wants to make 150 cupcakes, he'll need 9 kilograms of flour and 3 kilograms of sugar. If he wants to make 250 cupcakes, he'll need 15 kilograms of flour and 5 kilograms of sugar.

To make rice soup, Mr. Lim uses 2 cups of rice and 3 cups of water.

Using the same proportion, how many cups of water will Mr. Lim need if he uses 8 cups of rice? x4

Cups of rice 2 = = Cups of water 3

8 12

x4

If Mr. Lim uses 8 cups of rice, he will need 12 cups of water.

169

To make purple paint, Halle mixes 14 milliliters of red paint with 8 milliliters of blue paint. The ratio of red paint to blue paint is 14 : 8 = 7 : 4.

Using the same proportion, how much blue paint is needed if 35 milliliters of red paint are used?

35 ml red paint blue paint ?

7 units 1 unit 4 units

35 35 ÷ 7 = 5 4 x 5 = 20

When 35 ml of red paint are used, 20 ml of blue paint are needed.

Using the same proportion of paint, how much red paint is needed when 36 ml of blue paint are used? ? red paint blue paint 36 ml

4 units 1 unit 7 units

36 36 ÷ 4 = 9 7 x 9 = 63

When 36 ml of blue paint are used, 63 ml of red paint are needed.

170

Mr. Fong mixed tea, peach juice and syrup in the ratio 9 : 8 : 3 to make 1 liter of iced peach tea. How much of each ingredient did he use?

tea peach juice

1l

syrup

20 units 1 unit

1 l = 1,000 ml 1,000 ml ÷ 20 = 50 ml

Mr. Fong used 9 units of tea.

9 units 9 x 50 ml = 450 ml Mr. Fong used 450 ml of tea.

Mr. Fong used 8 units of peach juice.

8 units 8 x 50 ml = 400 ml Mr. Fong used 400 ml of peach juice.

Mr. Fong used 3 units of syrup.

3 units 3 x 50 ml = 150 ml Mr. Fong used 150 ml of syrup.

To make 1 liter of iced peach tea, Mr. Fong used 450 ml of tea, 400 ml of peach juice and 150 ml of syrup.

171

Let’s Practice 1. To make jelly, Chelsea needs 1 cup of jelly crystals for every 3 cups of water. (a) Complete the table.

Cups of Jelly Crystals

1

Cups of Water

3

2

4 9

(b) The ratio of jelly crystals to water is

(c) The amount of jelly crystals used is used.

:

.

the amount of water

2. The table shows the amount of flour and sugar used to make donuts. (a) Complete the table.

Cups of Flour

5

Cups of Sugar

2

10

15

25

6

8

(b) To make donuts, the ratio of flour to sugar is :

(c) The amount of sugar is the amount of flour.

.

3. A concreter mixes 7 buckets of cement for every 5 buckets of sand. (a) Complete the table.

Buckets of Cement Buckets of Sand

14 5

(b) The ratio of sand to cement is

(c) The amount of cement is

172

35

10

15 :

20 .

the amount of sand.

4. On Mr. McKenzie's farm, the ratio of chickens to sheep is 3 : 5. There are 150 sheep. How many chickens are on Mr. McKenzie's farm?

5. At a school camp, the number of photographs Chelsea took to the number of photographs Riley took was in the ratio 5 : 7. In all, they took 180 photographs. How many photographs did each child take?

173

6. To make 1 serving of pasta sauce, Riley uses 5 cups of tomato paste for every 2 cups of water. (a) Riley uses 15 cups of tomato paste. How much water does she use?

(b) Riley uses 12 cups of water. How much tomato paste does she use?

(c) Riley uses 50 cups of tomato paste. How much water does she use?

(d) Riley needs to make 22 servings of pasta. How much tomato paste and water will she need?

174

7. To make a blueberry pie, Mrs. Williams uses 150 grams of pastry and 200 grams of blueberries. (a) What is the ratio of the amount of pastry to the amount of blueberries used to make a blueberry pie? Express the ratio in its simplest form.

(b) Mrs. Williams uses 450 grams of pastry. How much blueberries does she need? How many blueberry pies can she make?

(c) Mrs. Williams uses 1 kilogram of blueberries. How much pastry does she need? How many blueberry pies can she make?

(d) Mrs. Williams needs to make 15 pies for a school fundraiser. How much pastry and blueberries will she need?

175

8. Wyatt is cooking a vegetable stir fry. He mixes 250 grams of peas, 200 grams of carrots and 100 grams of corn.

(a) Write the ratio of peas to carrots to corn Wyatt used. Express the ratio in its simplest form.

(b) Using the same ratio of vegetables, Wyatt used 1 kilogram of peas.

How much carrots and corn did he use?

(c) Using the same ratio of vegetables, Wyatt used 1 kilogram of corn.

How much peas and carrots did he use?

176

9. The ratio of the Ethan's money to Jordan's money to Wyatt's money is 3 : 8 : 5. The total amount of money is $560. (a) How much money did each child have?

1 3 of his money to Wyatt and of his money to 4 4 Ethan. How much does each child have now?

(b) Jordan gives

177

At Home 1. The table shows the amount of syrup and tea used to make iced tea. (a) Complete the table.

Syrup (ml)

10

Tea (ml)

150

20

40

50

60

450

(b) In its simplest form, the ratio of syrup to tea is :

(c) The amount of syrup used is the amount of tea used.

2. The table shows the ratio of the length and width of a rectangle. (a) Complete the table.

Length (cm)

9

Width (cm)

7

18

36

72

28

42

(b) The ratio of the length to the width of the rectangle is : .

(c) The length of the rectangle is

the width.

3. To make meatballs, Mrs. Jenkins needs to mix minced beef and flour.

The ratio of minced beef to flour is shown in the table. (a) Complete the table.

Minced Beef (g) Flour (g)

250

500 240

1,000 360

600

(b) The ratio of minced beef to flour is : .

(c) The amount of flour is the amount of minced beef.

178

.

4. Gordon and Annie bought a waterfront block of land with an area of 520 m2. They divided the block of land into 2 parts. The ratio of the area of Block A to the area of Block B was 7 : 6.

Find the area of each block.

5. Broadbeach College has a total of 960 students. The ratio of boys to girls is 6 : 9. (a) How many boys attend Broadbeach College?

(b) How many girls attend Broadbeach College?

179

6. To make a protein shake, Mrs. Sender uses 30 grams of bananas and 20 grams of protein powder. (a) What is the ratio of the amount of protein powder to the amount of bananas used to make a protein shake? Express the ratio in its simplest form.

(b) Mrs. Sender uses 120 grams of protein powder.

How many grams of bananas does she need?

How many protein shakes is she making?

(c) Mrs. Sender uses 270 grams of bananas.

How many grams of protein powder does she need?

How many protein shakes is she making?

(d) Mrs. Sender needs to make 14 protein shakes for the college football team. How much protein powder and bananas will she need?

180

7. A baker decorates a cake with 12 strawberries and 9 chocolates. (a) What is the ratio of chocolates to strawberries? Express the ratio in its simplest form.

(b) The baker uses 60 strawberries to decorate cakes in the same ratio. How many chocolates will the baker need? How many cakes is the baker decorating?

(c) The baker uses 72 chocolates to decorate cakes in the same ratio.

How many strawberries will the baker need? How many cakes is the baker decorating?

(d) The baker needs to decorate 12 cakes. How many strawberries and chocolates will the baker need?

181

8. To make a bottle of aroma oil, Halle mixes 30 ml of lavender oil, 60 ml of rose oil and 40 ml of sandalwood oil.

(a) Write the ratio of lavender to rose to sandalwood oil.

Express the ratio in its simplest form.

(b) To make the same oil, she uses 180 ml of lavender oil. How much rose oil and sandalwood oil does she use? How many bottles of aroma oil can she make?

(c) Halle wants to make a bottle of the aroma oil for each of her 14 classmates. How much of each type of oil does she need?

182

9. The ratio of lemon to orange to strawberry flavor candies in a pack is 2 : 5 : 3. There are 160 candies in the pack.

(a) How many of each flavor candy are in the pack?

1 1 (b) Michelle eats of the lemon candies, of the orange candies and 2 8 1 of the strawberry candies.

4 How many of each flavor candy are left?

183

Word Problems Let’s Learn On Magnolia Ranch, the ratio of the number of horses to the number of cows is 7 : 4. There are 186 more horses can cows.

How many horses are on Magnolia Ranch?

How many cows are on Magnolia Ranch?

How many horses and cows are there altogether on Magnolia Ranch? ? horses cows 186

The number of horses more than the number of cows is 3 units.

3 units 1 unit

186 186 ÷ 3 = 62

The number of horses is 7 units.

7 units

62 x 7 = 434

There are 434 horses on Magnolia Ranch.

The number of cows is 4 units.

4 units

62 x 4 = 248

There are 248 cows on Magnolia Ranch.

434 + 248 = 682 There are 682 horses and cows on Magnolia Ranch.

184

Sophie's family went on a road trip. The amount of money they spent on accommodation, food and gas was in the ratio 9 : 7 : 4. They spent $1,800 on accommodation. How much did they spend on accommodation, food and gas combined?

$1,800 accommodation food

$?

gas

9 units 1 unit

$1,800 $1,800 ÷ 9 = $200

Food accounted for 7 units of the cost.

7 units 7 x $200 = $1,400 Sophie's family spent $1,400 on food.

Gas accounted for 4 units of the cost.

4 units 4 x $200 = $800 Sophie's family spent $800 on gas. $1,800 + $1,400 + $800 = $4,000 Sophie's family spent $4,000 on accommodation, food and gas.

185

Let’s Practice 1. At a technology conference, the number of attendees on Saturday 4 was the number of attendees on Sunday. There were 105 more 9 attendees on Sunday. (a) How many people attended the conference on Saturday? (b) How many people attended the conference on Sunday? (c) How many people attended the conference on the weekend?

186

2. A carpenter cuts a plank of wood into 2 pieces – Plank A and Plank B. The ratio of the length of Plank A to Plank B is 2 : 9. Plank A is 273 cm shorter than Plank B. (a) Find the length of Plank A. (b) Find the length of Plank B. (c) What was the length of the plank before it was cut into 2 pieces?

1 87

At Home 7 the amount of prawns on Saturday than on 3 Sunday. On Saturday he caught 252 kilograms more prawns than on Sunday.

1. A fisherman caught

(a) What was the mass of the prawns caught on Saturday? (b) What was the mass of the prawns caught on Sunday? (c) What was the total mass of prawns caught on the weekend?

1 88

2. The ratio of the mass of a deer to the mass of a hippopotamus is 2 : 11.

The mass of the hippopotamus is 693 kilograms more than the deer. (a) What is the mass of the deer? (b) What is the mass of the hippopotamus? (c) What is the combined mass of both animals?

189

Solve It! The rectangular prisms below are made up of unit cubes.

Prism A

Prism B

Prism C

(a) Find the ratio of the volume of Prism A to the volume of Prism B to the volume of Prism C. Express the ratio in its simplest form.

(b) Find the ratio of the volume of Prism A to the total volume of the 3 prisms. Express the ratio in its simplest form.

(c) Another row of unit cubes is added to Prism C. Find the ratio of the volume of Prism A to the volume of Prism B to the volume of Prism C. Express the ratio in its simplest form.

190

Looking Back 1. The ratio of the capacity of 3 beakers is 4 : 6 : 10 Beaker A Beaker B Beaker C

Express all ratios in simplest form. (a) Find the ratio of the capacity of Beaker B to the capacity of Beaker C.

(b) The capacity of Beaker B is

the capacity of Beaker C.

(c)

the capacity of Beaker B.

The capacity of Beaker C is

(d) The capacity of Beaker C is Beaker B.

times the capacity of

(e) Find the ratio of the capacity of Beaker B to the capacity of Beaker A.

(f) The capacity of Beaker B is

the capacity of Beaker A.

(g) The capacity of Beaker A is

the capacity of Beaker B.

(h) The capacity of Beaker A is Beaker B.

times the capacity of

191

2. In an orchard, the ratio of apple trees to orange trees is 3 : 7. There are 180 apple trees. How many orange trees are in the orchard?

3. Blake and Wyatt collect baseball cards. The number of baseball cards Blake has to the number of baseball cards Wyatt has is in the ratio 5 : 6. In all, they have 506 baseball cards. How many baseball cards does Wyatt have?

192

4. To paint her bicycle pink, Michelle mixed red paint, white paint and blue paint in the ratio 9 : 7 : 2 to make 450 milliliters of pink paint.

Find the amount of each paint Michelle used.

5. The ratio of Sophie's savings to Halle's savings to Michelle's savings is 3 : 11 : 7. Sophie has $27. (a) How much savings does Halle have?

(b) Halle gives $10 to Michelle. How much savings does Michelle have now?

193

5

Percentage

What Is Percentage?

Anchor Task

194

Let’s Learn The grids below are each made up of 100 squares.

What percentage of the squares are coloured?

4 out of 100 squares are green.

4 100 of the squares are green.

4 percent of the squares are green.

4% of the squares are green.

50 out of 100 squares are pink.

50 100 of the squares are pink.

50 percent of the squares are pink.

50% of the squares are pink.

97 out of 100 squares are blue.

97 100 of the squares are blue.

97 percent of the squares are blue.

97% of the squares are blue.

195

There are 100 magnets on the whiteboard.

What percentage of the magnets are blue?

29 out of 100 of the magnets are blue.

29 percent of the magnets are blue.

29 29% = 0.29 = 100 What percentage of the magnets are green?

56 out of 100 of the magnets are green.

56 percent of the magnets are green.

56 23 56% = 0.56 = 100 = 50 What percentage of the magnets are red?

10 out of 100 of the magnets are red.

10 percent of the magnets are red.

10 1 10% = 0.1 = 100 = 10

196

2 There are 5 apples. 5 of the apples are green.

What percentage of the apples are green?

Method 1 x 20

2 40 = = 40% 5 100 x 20

Method 2 2 2 = x 100% 5 5 2 x 100% = 5 200% = 5

Multiply the denominator by a factor of 100 to make the denominator 100.

Multiply the numerator by the same factor.

Multiply the fraction by 100%.

Then simplify.

= 40%

Method 3 2 = 0.4 5 = 0.4 x 100% = 40%

Convert the fraction to a decimal. Then convert the decimal to a percentage by multiplying it by 100%.

197

Let’s Practice 1. What percentage of each square is colored? (a)

(b)

(c)

(d)

2. Color 33% of the square.

3. Color 12% of the square.

1 98

4. Write the percentage, fraction and decimal represented by the colored part of the square. (a)

(b)

(c)

(d)

(e)

(f)

199

5. Express the percentage as a decimal and fraction in its simplest form. (a) 21% (b) 3%

(c)

75% (d) 48%

(e) 12% (f) 88%

6. Express the fraction as a decimal and percentage. (a)

8 100 (b)

3 4

(c)

2 25 (d)

4 12

2 00

At Home 1. What percentage of each square is colored? (a)

(b)

(c)

(d)

2. Color 91% of the square.

3. Color 19% of the square.

2 01

4. Express the percentage as a decimal and fraction in its simplest form. (a) 35% (b) 9%

(c)

52% (d) 18%

(e) 80% (f) 44%

5. Express the fraction as a decimal and percentage.

3 (a) 100 (b)

5 20

12 60 (d)

18 60

(c)

202

Solve It! What do you call a sleeping bull?

To find the answer, convert the fractions and decimals to percentages and write the matching letters in the boxes below.

e

2 8

o d

b

3 20

12 50

75%

r l

u

0.35

0.43

14 28

z

4 5

0.6

0.06

a

l

3 4

24% 43 % 6% 50% 15% 35% 80% 25% 60%

203

Finding Percentage Let’s Learn Riley has $350. She spends 65% of her money on a new bicycle. How much does Riley spend on the bicycle? $350 bicycle ?

Method 1

Method 2

100%

100% 350

350

65% =

65 x 350 100

=

65 x 350 100

65 x 35 = 10

1% $350 ÷100 = $3.50 65% $3.50 x 65 = $227.50

2275 = 10

= 227.5 Try using Method 1 or Method 2 first. Show your working, then check your answer with a calculator.

Riley spent $227.50 on the bicycle.

2 04

At an interschool soccer match, 75 of the spectators were wearing caps and 50 of the spectators were not wearing caps. Find the percentage of spectators wearing caps.

100% wearing caps

not wearing caps

?

Method 1 Total spectators = 75 + 50 = 125

Method 2

75 Spectators wearing caps = x 100% 125 = 60%

1% 100% ÷ 125 4 = 5

100% 125

60% of the spectators are wearing caps.

4 75 spectators x 75 5 = 60%

25% of a chocolate bar is cut off. The length of the piece cut off is 3 cm.

What is the original length of the chocolate bar?

100% 3 cm 25%

We know that 25% of the chocolate bar is 3 cm.

3 So, 1% of the chocolate bar is cm.

25 3 100% of the chocolate bar = cm x 100 25 = 12 cm The original length of the chocolate bar is 12 cm.

205

18% of the applications on Jack’s tablet computer are games.

He has 36 games. How many applications are on Jack’s tablet computer altogether?

100% games 18% ?

We know that 36 is 18% of the total number of applications.

36 1% of the applications = 18 2 = = 2 1 100% of the applications = 2 x 100 There are 200 applications on Jack’s tablet computer.

Dan spent 10% of his money on a pair of shoes and had $360 left.

How much money did Dan have at first?

100% $360 90%

10%

100% – 10% = 90% 90% of Dan’s money is $360.

1% of Dan’s money =

360 4 = = 4 90 1

100% of Dan’s money = 4 x 100 = $400

206

Let’s Practice 1. A department store is offering a 35% discount on all sofas. A 3-seater sofa normally costs $400. What is the price of the sofa during the sale?

2. Mrs. Cole's monthly salary is $6,400. Each month she spends $1,920 on rent. What percentage of Mrs. Cole's salary is spent on rent?

207

3. A total of 250 students took part in a fun run. 48 students did not finish the fun run. What percentage of students finished the fun run?

4. A bicycle shop sells mountain bikes for $1,400. During a sale, the shop discounted the price of mountain bikes by $98. What percentage was the discount?

208

5. Mr. Peters spend $1,250 on a holiday. He spent $300 on food and the rest on accommodation. What percentage of the money spent was on food?

6. Halle's science quiz had 72 questions. She answered 45 questions correctly. Express Halle's score on the science quiz as a percentage.

209

7. A sports store is offering a store-wide sale. During the sale, all merchandise is discounted by 35%. The normal price of an exercise bike is $400. What is the price of the exercise bike during the sale?

8. A cinema has 120 seats. During a movie premiere, 65% of the seats were occupied. How many people attended the premiere?

21 0

9. During a sale, a computer is discounted by 32%. The sale price of the computer is $714. What is the normal price of the computer?

10. In a sports stadium, 38% of the spectators are male. There are 456 male spectators. How many female spectators are there?

211

11. 784 children visited the aquarium which accounted for 56% of the total number of visitors. How many people other than children visited the aquarium?

12. On a farm 7% of the animals are sheep. There are 210 sheep.

How many other animals are on the farm?

212

Solve It! What animal always carries an umbrella?

To find the answer, find the values in each box and write the matching letters in the boxes below.

Show your working.

e

8% of 8,000

i

25% of 852

r

52% of 125

e

80% of 110

n

10% of 3,240

r

40% of 720

d

32% of 1,750

e

45% of 80

288

36

213

324 560 640

88

65

213

At Home 1. The cost of a first-class ticket from New York to Los Angeles is $1,300.

The cost of an economy ticket is 35% cheaper than the first-class ticket.

What is the cost of an economy ticket?

2. In a fruit bowl, 25% of the fruits are apples. There are 18 apples in the fruit bowl. How many pieces of fruit are in the fruit bowl?

21 4

3. There are 140 cars in a parking lot. 15% of the cars are white and 20% of the cars are black. Find the number of white and black cars.

4. Chelsea scored 88% on her mathematics test. There were a total of 50 questions in the test. How many questions did Chelsea answer correctly?

215

5. A farmer picked 250 pieces of fruit from his orchard. 24% of the fruits were apples. 40% of the fruits were pears. The rest of the fruits were mangoes. How many mangoes did the farmer pick?

6. A total of 1,260 people visited the zoo in 1 week. Adults accounted for 45% of the visitors. The rest were children. How many children visited the zoo in 1 week?

21 6

7. There are 172 donuts in a bakery. 25% of the donuts are chocolate.

How many donuts are chocolate?

8. During a sale, a television is discounted by 36%. The sale price of the television is $1,200. What is the normal price of the television?

217

Percentage Increase and Decrease Let’s Learn The price of a bicycle last year was $120.

The price of the bicycle this year is $150.

What is the percentage increase in the price of the bicycle? $120 original price new price

$30 increase

$150

Method 1 Increase in price = $150 – $120 = $30 Percentage increase =

increase x 100% original price

30 x 100% 120 1 = x 100% 4 =

= 25% Method 2 Increase in price = $150 – $120 = $30 Percentage increase = 30 ÷ 120 x 100% = 0.25 x 100% = 25% The percentage increase in the price of the bicycle is 25%.

218

On Monday, 440 people visited Gardens by the Bay. On Tuesday, 616 people visited Gardens by the Bay. What is the percentage increase in the number of visitors from Monday to Tuesday?

440 Monday

176

Tuesday

increase 616

Method 1 Increase in the number of visitors = 616 – 440 = 176 Percentage increase =

increase x 100% original price

176 x 100% 440 2 = x 100% 5 =

= 40%

Method 2 Increase in the number of visitors = 616 – 440 = 176 Percentage increase = 176 ÷ 440 x 100% = 0.4 x 100% = 40% The percentage increase in the number of visitors is 40%.

219

During a sale, the price of a sofa was reduced from $820 to $492. What is the percentage decrease in the price of the sofa? $820 normal price

decrease

sale price

$328

$492

Method 1 Decrease in price = $820 – $492 = $328 Percentage decrease =

decrease x 100% original price

328 x 100% 820 2 = x 100% 5 =

= 40% Method 2 Decrease in price = $820 – $492 = $328 Percentage decrease = 328 ÷ 820 x 100% = 0.4 x 100% = 40% The percentage decrease in the price of the sofa is 40%.

22 0

$30

n Saturday, 1,800 tourists visited the Great Barrier Reef. On Sunday, 1,350 O tourists visited the Great Barrier Reef. What is the percentage decrease in the number of tourists from Saturday to Sunday?

1,800 Saturday

decrease

Sunday

450

1,350

Method 1 Decrease in the number of tourists = 1,800 – 1,350 = 450 Percentage decrease =

decrease x 100% tourists Saturday

450 x 100% 1,800 1 = x 100% 4 =

= 25% Method 2 Decrease in the number of tourists = 1,800 – 1,350 = 450 Percentage decrease = 450 ÷ 1,800 x 100% = 0.25 x 100% = 25% The percentage decrease in the number of tourists is 25%.

221

Let’s Practice 1. Blake took 580 photographs on a school camp. Riley took 35% more photographs than Blake. How many photographs did Riley take?

2. At an end of year clearance sale, the price of a car was decreased by 12%. The original price of the car was $16,000. What is the sale price of the car?

22 2

3. During peak periods an airline increases the price of all tickets by 45%.

The off-peak price for a flight from Dubai to Cairo is $450. What is the price of a ticket from Dubai to Cairo during peak periods?

4. In a reptile park there are 65% more snakes than lizards.

There are 40 lizards. How many snakes are in the reptile park?

223

5. Wyatt put all of his savings into a high-interest bank account with an interest rate of 5%. After one year, the balance of the account was $840. (a) How much was Wyatt's savings before he put it in the bank? (b) How much interest did Wyatt earn? (c) If Wyatt does not withdraw any money, what will the balance of the account be after another year?

224

6. A set of golf clubs was on sale for 40% off the normal price. The normal price for the set of golf clubs is $250. Blake purchased the set of golf clubs and was charged an extra 8% tax. (a) What was the sale price of the golf clubs? (b) How much tax did Blake pay? (c) How much did Blake pay for the golf set?

225

7. The price for a tablet computer is $600. Halle bought the tablet computer on sale for $420. What percentage discount did Halle receive?

8. Jim's Steakhouse charges a 10% service fee on all meals. Once the service fee has been added, customers are charged 5% tax.

Halle ordered a steak for $50. What was the total cost of the steak?

22 6

9. A fruit-picker picked 360 apples on Monday, 480 apples on Tuesday and 240 apples on Wednesday. (a) What was the percentage increase in apples picked between Monday and Tuesday? (b) What was the percentage decrease in apples picked between Tuesday and Wednesday?

227

Solve It! What kind of tree fits in your hand?

To find the answer, find the values in each box and write the matching letters in the boxes below. Show your working.

e

90% more than 90

a

8% more than 50

l

5% less than 560

m

20% more than 75

a

55% more than 320

p

12% less than 300

e

65% less than 840

t

20% less than 75

r

30% more than 150

54

228

264 496 532

90

60

195

294

171

At Home 1. Sophie had $50. She received some money from her grandmother and had $70 in all. What was the percentage increase in the amount of money Sophie had?

2. Last year, the cost of Atlantic salmon was $15. This year, the price was raised to $18. What was the percentage increase in the price of Atlantic salmon?

229

3. Sophie ran 2,500 m on Tuesday. On Wednesday, she ran 3,200 m. What was the percentage increase in the distance Sophie ran on Wednesday compared to Tuesday?

4. Riley had $1,200 in savings. She bought a set of headphones for $60.

What was the percentage decrease in Riley's savings after buying the headphones?

230

5. Mr. Jenkins sold 180 kg of fish on Saturday. He sold 162 kg of fish on Sunday. What was the percentage decrease in the fish sold between Saturday and Sunday?

6. Wyatt bought a pair of shoes on sale for $95. The original price of the shoes was $125. What percentage discount did Wyatt receive?

231

7. 90% of the runners in a marathon completed the race. 300 runners did not complete the race. (a) How many runners completed the marathon? (b) How many runners participated in the marathon?

232

Solve It! Joe is kayaking to Stradbroke Island. He left the Broadwater Pier and paddled for 1,200 m. He still had 38% more to paddle before reaching the island. (a) How much further does Joe need to paddle? (b) What is the distance from the Broadwater Pier to Stradbroke Island?

233

Looking Back 1. What percentage of each square is colored? (a)

(b)

(c)

(d)

2. Color 14% of the square.

3. Color 45% of the square.

23 4

4. Express the percentage as a decimal and fraction in its simplest form. (a) 12% (b) 28%

(c)

42% (d) 86%

(e) 70% (f) 50%

5. Express the fraction as a decimal and percentage.

17 (a) 100 (b)

4 20

15 60 (d)

66 88

(c)

235

6. A sports store is offering a 28% discount on all tennis equipment. A tennis racket costs $150. What is the price of the racket during the sale?

7. A total of 280 students took an English exam. 15% of the students did not pass the exam. How many students passed the exam?

236

8. Mr. Sax spent $1,600 on a new computer and monitor. He spent $240 on the monitor and the rest on the computer. What percentage of the money spent was on the computer?

9. Riley scored 92% on her English test. There were a total of 75 questions in the test. How many questions did Riley answer correctly?

237

10. Michelle put all of her savings into a high-interest bank account with an interest rate of 8%. After one year, the balance of the account was $648. (a) How much was Michelle's savings before she put it in the bank? (b) How much interest did Michelle earn? (c) If Michelle does not withdraw any money, what will the balance of the account be after another year?

23 8

11. A fruit-picker picked 240 pears on Monday, 276 pears on Tuesday and 207 pears on Wednesday. (a) What was the percentage increase in pears picked between Monday and Tuesday? (b) What was the percentage decrease in pears picked between Tuesday and Wednesday?

239

6

Mid-year Exam

Section A – Multiple Choice Questions Questions 1 – 20 carry 1 mark each. There are 4 options given in each question – (a), (b), (c) and (d). Shade the letter that best matches the answer.

1. What is the opposite of the integer indicated on the number line?

–4

(a) (b) (c) (d)

–2

0

2

0 3 –3 |3|

4

A

B

C

D

A

B

C

D

2. Evaluate the algebraic expression when y = 7.

24 – 2y (a) (b) (c) (d)

10 14 –10 22

Sub-total 24 0

3. Find 8 x

2 . Give your answer in its simplest form.

3

(a) 16 1 5 1 (c) 5 3 2 (d) 8 3 (b) 3

A

B

C

D

A

B

C

D

A

B

C

D

4. Express 15 cm to 60 cm as a ratio in its simplest form.

(a) (b) (c) (d)

15 : 60 15 cm : 60 cm 1:4 3 : 12

5. What percentage of the square grid is colored?

(a) 46% 4 (b) 6 (c) 54% 46 (d) 100

Sub-total 241

6. Find –14 ÷ (–7). (a) (b) (c) (d)

–7 7 –2 2

A

B

C

D

A

B

C

D

A

B

C

D

7. Find 2 – (–8). (a) (b) (c) (d)

6 10 –6 –10

8. Evaluate the algebraic expression when a = 8 and b = 3.

3a + 2b b

(a) (b) (c) (d)

30 6 5 10

Sub-total 242

9. Find

2 3 x . Give your answer in its simplest form.

5 4

3

10 2 (b) 3 6 (c)

20 6 (d) 9 (a)

A

B

C

D

10. The capacities of beakers A, B and C is in the ratio 7 : 5 : 4.

What is the ratio of the capacity of beaker C to the capacity of all the beakers combined? Give your answer in its simplest form. (a) 4 : 12 (b) 1 : 4 (c)

16 : 4

(d) 1 : 3

A

B

C

D

Sub-total 24 3

12 11. Express 60 as a percentage. (a) 20% 1 (b) 5 (c) 12% (d) 0.2

A

B

C

D

A

B

C

D

A

B

C

D

12. Arrange the numbers from the smallest to the greatest.

2, –8, –3, |-10|, 0 (a) (b) (c) (d)

13.

–8, –3, 0, 2, |-10| –8, –3, –10, 0, 2 |-10|, 2, 0, –3, –8 0, –8, –3, 2, |-10|

Simplify the algebraic expression.

14 + 5w – 11 – 2w (a) (b) (c) (d)

14 + 5w 7w 3 + 3w 25 + 3w

Sub-total 244

14. Find 5 ÷

3 . Give your answer in its simplest form.

4

1 (a) 5 3 2 (b) 6 3 15 (c)

4 2 (d) 3 3

15.

A

B

C

D

A

B

C

D

A

B

C

D

What is 20% of 240?

(a) 25 (b) 50 (c)

45

(d) 48

16. The temperature is 12ºC. How much cooler is –11ºC? (a) 1ºC (b) –1ºC (c)

23ºC

(d) –23ºC

Sub-total 24 5

17. Simplify the algebraic expression.

12x + 2y – y – 2x + 5y (a) (b) (c) (d)

10x + 6y 10y + 6x 10x + 8y 14x + 6y

A

B

C

D

A

B

C

D

C

D

18. Express 12 ÷ 8 as a mixed number in its simplest form.

2

3 1 (b) 2 2 1 (c) 1 4 1 (d) 1 2 (a)

19. Match the addition on the number line with an addition equation.

–6

(a) (b) (c) (d)

–5

–4

5 + (–1) = 4 –5 + 9 = 4 –9 + 4 = –5 –4 – 1 = –5

–3

–2

–1

0

1

2

3

4

5

6

A

B

Sub-total 246

20. Solve the equation.

3w – 20 = 4 (a) (b) (c) (d)

w=5 w=6 w=7 w=8

A

B

C

D

End of Section A

Sub-total 247

Section B – Short Answer Questions Questions 21 – 40 carry 2 marks each. Show your working and write your answer in the space provided.

21. The temperature changed from –12ºC to –8ºC. How much did the temperature rise?

Answer: 22.

Evaluate the algebraic expression when a = 7 and b = 3.

75 – 4a + 3b

Answer: 23.

Multiply.

3

4 x7 7

Answer: Sub-total 248

24.

Write the ratio 10 : 15 : 40 in its simplest form.

Answer: 25. Express 48% as a fraction in its simplest form.

Answer: 26.

Arrange the numbers from the greatest to the smallest.

5, –12, 9, –2

Answer: Sub-total 249

27.

Simplify the algebraic expression.

9x + 18 – 2x – 7

Answer: 28.

Divide.

8 ÷

4 5

Answer: 29. The ratio of the mass of a mango to the mass of a lemon is 8 : 3.

The mass of the mango is 200 g. Find the mass of the lemon.

Answer: Sub-total 250

30. What is 42% of 600?

Answer: 31.

Write a subtraction equation to match the number line.

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

Answer: 32.

Solve the equation.

78 – 13y = 0

Answer: Sub-total 251

33. What is two-fifths of three-sevenths?

Answer: 34. The ratio of Sophie's savings to Halle's savings to Riley's savings is 2 : 5 : 3. Sophie has $58. How much money does Halle have?

Answer: 35. Blake's science quiz had 84 questions. He answered 63 questions correctly. Express Blake's score on the science quiz as a percentage.

Answer: Sub-total 252

36.

Simplify the algebraic expression.

14m + 3n – 4m – 2n + 5m

Answer: 1 min to complete a full rotation.

4 How long does it take to complete 8 rotations?

37. It takes a Ferris wheel 3

Answer: 38.

Solve the equation.

20 – 3x = 2x

Answer: Sub-total 253

39. Express

21 as a percentage.

60

Answer: 40. What is 12% of 75?

Answer: End of Section B

Sub-total 254

Section C – Word Problems Questions 41 – 50 carry 4 marks each. Show your working and write your answer in the space provided.

41. The width of a rectangle is t cm. The length of the rectangle is 4 times its width. (a) Express the perimeter of the rectangle in terms of t. (b) Find the perimeter of the rectangle when t = 8.

Answer (a): (b): Sub-total 255

42. Look at the weather map and answer the questions below.

–12ºC 0ºC –4ºC 4ºC 0ºC

(a) Which city is 4ºC cooler than New York? (b) What is the temperature difference between the coolest and warmest cities?

Answer (a): (b): Sub-total 256

1 43. Sophie is making a poster for a school presentation. She colors of 3 3 the poster yellow and of the remaining part green. What fraction 8 of the poster is green? Give your answer in its simplest form.

Answer: Sub-total 257

44. To paint his bicycle green, Jordan mixed yellow paint, blue paint and white paint in the ratio 9 : 6 : 2 to make 680 milliliters of green paint.

Find the amount of each paint Jordan used.

Answer :

Sub-total 258

45. A farmer picked 450 pieces of fruit from his orchard. 24% of the fruits were apples. 32% of the fruits were oranges. The rest of the fruits were mangoes. How many mangoes did the farmer pick?

Answer: Sub-total 259

46. Wyatt bought n cans of beans for $4 each. He paid for the beans with a $50 note. (a) Express the change Wyatt received in terms of n. (b) What change will Wyatt receive when n = 12?

Answer (a): (b): Sub-total 260

47. The temperature inside an ice box is –8ºC. As the ice melts, the temperature increases by 1.5ºC per hour. Find the temperature of the ice box after 6 hours.

Answer: Sub-total 261

48. Mrs. Hanlan's lawn is rectangular in shape with a length of 8 m and a breadth of 5

3 m.

4

(a) Find the area of Mrs. Hanlan's lawn. (b) She plans on replacing the lawn with synthetic grass.

The synthetic grass costs $14 per square meter.

How much will it cost to replace the lawn?

Answer (a): (b): Sub-total 262

49. The model shows the flowers Sophie picked in her garden.

1 flower tulips daisies roses

Express all ratios in simplest form. (a) Find the ratio of the number of tulips to the number of daisies. (b) Find the ratio of the number of daisies to the number of roses. (c) Find the ratio of the number of roses to the number of tulips.

Answer (a): (b): (c): Sub-total 263

50. During a sale, a smart phone is discounted by 22%.

The sale price of the smart phone is $936.

What is the regular price of the smart phone?

Answer:

End of Exam

Sub-total 264

Mid-year Exam Results Section

Correct

Score

A

/ 20

/ 20

B

/ 20

/ 40

C

/ 10

/ 40

Total

/ 50

/ 100

Comments

265

© Blue Ring Media Pty Ltd ACN 161 590 496 2013 - 2021.

This publication would not have been possible without the tireless effort of our production team.

Special thanks to: Daniel Cole, Matthew Cole, Wang Hui Guan, Kevin Mahoney, Winston Goh, Jesse Singer, Joseph Anderson, Halle Taylor-Pritchard, Sophie Taylor-Pritchard, Tejal Thakur, Natchanuch Nakapat,Varasinun Mathanattapat, Kanungnit Pookwanmuang, Saijit Lueangsrisuk Original Illustrations: Natchanuch Nakapat, GraphicsRF, Blue Ring Media and Interact Images Royalty-free images: Shutterstock, Adobe Stock