pÄ€ngarau

mathematics

MX1011 understanding decimals, fraction and percentages ncea level 1

2011/3

mathematics ncea level 1

Expected time to complete work This work will take you about eight hours to complete. You will work towards the following standard: Achievement Standard 91026 (Version 1) Apply numeric reasoning when solving problems Level 1, Internal 4 credits In this booklet you will focus on these learning outcomes: • solving problems using decimals • solving problems using fractions • solving problems using percentages • solving problems using a combination of decimals, fractions and percentages. You will continue to work towards this standard in booklets MX1012 and MX1013.

Copyright © 2011 Board of Trustees of Te Aho o Te Kura Pounamu, Private Bag 39992, Wellington Mail Centre, Lower Hutt 5045, New Zealand. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means without the written permission of Te Aho o Te Kura Pounamu.

© te ah o o te k u ra p ou n am u

contents 1 Decimals 2 Understanding fractions 3

Solving problems with fractions

4 Percentages 5

Discounts, payments and GST

6

Percentage increases and decreases

7

Review activity

8 Answer guide

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how to do the work When you see:

1A

Complete the activity.

Check your answers.

Your teacher will assess this work.

Contact your learning advisor.

Check the website.

Complete the online activity.

You will need: • a pen, paper and a scientific calculator • access to the Internet (helpful, but not essential). Resource overview Write your working and your answers in the spaces provided.

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decimals learning outcome Solve problems using decimals.

learning intentions In this lesson you will learn to: • understand the meaning of place value • apply your knowledge of decimals • solve problems containing decimals.

introduction The aim of this booklet is for you to revise your understanding of straightforward problems using decimals, fractions and percentages. There is also an opportunity to learn how to solve long problems that involve sorting information. If you find the work in this booklet too easy, go straight to the assessment task at the end of the booklet, complete the assessment and return it to your teacher. Then start on booklet MX1012. If you find the work in this booklet too difficult or you are having trouble with any part of it, please contact your mathematics and statistics teacher.

checking on place value

At the championships, Tamati ran the 100 metres race in 13.64 seconds.

His friend Jim ran the race in 13.7 seconds. Was he faster than Tamati? Is 13.64 bigger or smaller than 13.7?

It helps to put these numbers on a number line.

13.64 ↑ 13

13.6

13.7

14

The fastest person has the shortest time.

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In Tamatiâ€™s race, there were six runners. The time (in seconds) for each runner is given in the table below. AARON

BEN

JIM

SIONE

TAMATI

ZANE

13.02

12.08

13.7

11.3

13.64

11.57

Who was the fastest? Who had the quickest time? It is helpful to put these times in order from smallest to biggest.

1A

SIONE

ZANE

BEN

AARON

TAMATI

JIM

11.3

11.57

12.08

13.02

13.64

13.7

So, Sione won the race because he finished in the shortest amount of time. For each pair of numbers, circle the larger number. 1. 3.6 and 3.7 2. 63.7 and 63.69 3. 1.125 and 1.15 4. 0.701 and 0.71 5. 0.49 and 0.9 6. 3.002 and 3.2 7. Write down a number between 3.7 and 3.8. 8. Five competitors in the high jump recorded the following jumps, in metres. 1.32

4

1.29

1.26

1.03

1.3

1.19

Arrange these jumps in order from largest to smallest.

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decimals

9. a. The times for Heat 1 of the 100 metres girlsâ€™ freestyle swimming race at the championships were recorded, in seconds, as: 60.063

61.096

58.205

59.003

58.962

59.468

61.004

Write down the third fastest time. b. In Heat 2 of the 100 metres girlsâ€™ freestyle swimming race, a competitor recorded a time three 100ths of a second faster than the winner of Heat 1.

Write down the winning time for Heat 2.

Check your answers.

rounding Numbers do not always need to be expressed accurately. You can use rounding. Rounding a number means approximating a number to a less accurate number. (More on rounding will be covered in the next booklet, MX1012.) It is useful to know how to round to a given number of decimal places when solving problems containing decimals and fractions. The first decimal place.

Example 1 Round 9.23 to 1 decimal place (1 d.p.). Find the digit you are rounding to

9.23

If the next digit is 4 or smaller, leave the digit you are rounding to unchanged. 9.23 rounds to 9.2 (1 d.p.). Notice that on a number line, 9.23 is closer to 9.2 than it is to 9.3.

9.2

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9.23

9.3

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Example 2 Round 4.66666 to 1 decimal place. The first decimal place. The next digit after the first decimal place is 6, which is greater than 4. 4.66666 rounds to 4.7 (1 d.p.). Notice that on the number line, 4.66666 is closer to 4.7 than it is to 4.6.

4.6

4.66666

4.7

Example 3 Round 5.875 to 2 decimal places (2 d.p.). The second decimal place.

The digit after the second decimal place is 5, which is greater than 4. 5.875 rounds to 5.88 (2 d.p.).

1B

Round the following numbers as shown in the brackets. 1. 6.58 (round to 1 d.p.) 2. 3.4446 (round to 1 d.p.) 3. 0.056 (round to 1 d.p.) 4. 1.099 (round to 1 d.p.) 5. 16.99 (round to 1 d.p.) 6. 2.335 (round to 2 d.p.) 7. 20.33333 (round to 2 d.p.) 8. 12.181818 (round to 2 d.p.) Check your answers.

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decimals

solving word problems When solving problems, sometimes the hardest part is figuring out what the question is asking you to do. For each of the questions below, think about how to solve the problem. Will you need to add, subtract, multiply or divide? Or do a combination of these? In each space below, write down which of these operations you would use to solve the problem: add, subtract, multiply or divide.

vanessaâ€™s sewing project Vanessa bought 2.4 metres of fabric at $14.50 per metre. How much does she need to pay for the fabric? To solve this problem, I need to

Vanessa uses 1.86 metres of the fabric to make a skirt. How much fabric does she have left? To solve this problem, I need to

Vanessa also bought some thread, which cost $2.30, and a zip, which cost $1.65. How much did she spend altogether? To solve this problem, I need to

Vanessa had planned to spend up to $40 on the skirt. How much money does she have left? To solve this problem, I need to

Vanessa is going to cut the piece of fabric that is left over into three strips of equal width. How wide is each strip? To solve this problem, I need to Check your answers.

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simon's garage Simon is building a garage.

He has a length of timber 3.200 metres long and needs to cut 5 joists, each 466 millimetres long.

What length of timber will be left over? Ignore any timber lost in the process of making the cut.

You need to be careful about units: 466 mm = 0.466 m.

1m = 1 000 mm 1 mm = 0.001 m 466 mm = 0.466 m

One joist is 0.466 m long. Five joists’ total length: 5 × 0.466 = 2.33 m Timber left over: 3.200 – 2.33 = 0.870 m In this problem, there is a combination of multiplication and subtraction. Many different words and expressions can mean the same thing. There are many words in English that we use instead of add, subtract, multiply and divide. Here are some examples. Add:

Sum, plus, altogether, total, more than

Subtract: Minus, take away, less than, the difference between, amount left over Multiply: Times, lots of Divide:

8

Share amongst, divide between

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1C

In each of the problems below, underline the words that tell you what to do and then write down the type of problem: addition, subtraction, multiplication or division. Then solve the problem, showing your working. 1. The length of ribbon needed for a dress is 1.06 metres less than 2.83 metres. Find the length of ribbon needed.

2. The price of tomatoes was $5.45 per kilo last week; this week, it has increased by $1.35 per kilo. Find the price of tomatoes this week.

3. Hemi has 5.6 kilograms of fish to share amongst 8 friends. How much fish does each friend get?

4. Johnny picked 6.55 kilograms more apples than Joe, who picked 35.7 kilograms. Find how many kilograms of apples Johnny picked.

5. Sue ran 1.6 kilometres farther than Jane, who ran 6.82 kilometres. How far did Sue run?

6. The population of a city has fallen by 26 500 during the last decade. If the population was 126 350 a decade ago, what is it now?

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7. Jason earned $35.40 per week for 8 weeks. How much money did he earn in total?

8. If I am 1.62 metres tall and you are 1.85 metres tall, what is the difference in our heights?

9. Sonia bought 2 muffins at $1.40 each and a banana for 70 c. How much change does she get from $5.00?

10. There were 8 pizzas to share amongst 12 students. How much pizza does each student get?

Check your answers.

problem solving Some problems include lots of information. You need to read problems like this more than once. Think about how you can break down all the information into separate parts to help you to solve the problem. Here is an example. the basketball trip A basketball team is planning a trip from Nelson to Palmerston North for a tournament. You are asked to find the total cost for the trip and the cost per person.

They will take the school minibus with 10 players and 2 adults. Players and adults will pay the same amount.

It is 112 kilometres from Nelson to Picton and 145 kmilometres from Wellington to Palmerston North. The school charges 40 cents per kilometre to use the bus. Petrol will cost $1.80 per litre and an estimated 60 litres of petrol is needed for the trip. The ferry across Cook Strait will cost $420.00 each way, including the cost for the minibus. 10

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To get started on this question, make a list of all the different costs:

• school charge for the minibus

• petrol

• ferry.

Other factors to be considered to work out costs:

• number of people

• total distance travelled.

Put the information together in an organised way. costs

other factors

calculations

School charge for minibus $0.40 cents per km

Number of kilometres Nelson to Picton = 112 km Wellington to Palmerston North = 145 km

Cost of minibus 0.4 x 514 = $205.60

Total = 257 km Both ways = 2 × 257 = 514 km Petrol $1.80 per litre

Amount needed 60 L

Cost of petrol 1.80 × 60 = $108.00

Ferry $420 each way

Cost of the ferry $840.00

TOTAL COST

$1 153.60

Answer the question. What is the cost per person? There are 12 people altogether. $1 153.60 ÷ 12 = $96.13 (rounded to 2 d.p.). The cost per person is $96.13. Look at the process again.

• Make a list.

• Consider all factors.

• Organise the information.

• Do the necessary calculations (show your working).

• Answer the question, usually with a sentence.

This process is useful for solving problems with lots of information. Look at another example. © te aho o t e k ur a p o un a m u

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planning a barbeque A group of netball players is organising an end-of-season barbeque for all players in the club. You are asked to find the total cost of the barbeque. They expect around 180 people to turn up. The club will pay for the costs of the barbeque up to a maximum of $500. The group organising the barbeque is looking at the overall cost and they want to see if $500 is enough. They estimate 1 kilogram of sausages for every 5 people. Sausages on special will cost $8.90 per kilogram. They will buy 1 loaf of bread for every 10 people, at a cost of $3.35 per loaf, and 5 bottles of tomato sauce at $5.39 each. The cost of hiring the barbeque is $40. They will provide 10 bottles of soft drink at $1.90 each and 6 bottles of juice at $3.49 each. Club members will bring a salad to share, so that will not be part of the cost. Make a list of the costs:

• sausages

• bread

• tomato sauce

• barbeque hire

• drinks.

Consider other factors:

• weight of sausages to buy

• number of loaves of bread to buy.

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Organise the information. COSTS

OTHER FACTORS

CALCULATIONS

Sausages $8.90 per kilo

180 people 1 kg per 5 people 180 ÷ 5 = 36 kg

8.90 × 36 = $320.40

Bread $3.35 per loaf

180 people 1 loaf per 10 people 180 ÷ 10 = 18 loaves

3.35 × 18 = $60.30

Tomato sauce $5.39 per bottle

5 bottles

5 × 5.39 = $26.95

Barbeque hire $40.00

$40.00

Drinks Soft drink = $1.90 per bottle Juice = $3.49 per bottle

10 soft drink bottles 6 juice bottles 10 × 1.90 + 6 × 3.49

TOTAL COST

$39.94

$487.59

Answer the question. The total cost of the barbeque is estimated to be $487.59, which is less than the $500 budgeted.

1D

1. A group of 6 friends plan a holiday at the Kaiteriteri camping ground.

Work out the total cost of the holiday and how much it will cost each person.

The costs are to include petrol, accommodation and food.

The cabin will cost $60 for 2 people, plus $15 for each extra person each night. They plan to stay for 6 nights.

They each put in $80 to cover the cost of shared meals.

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They will drive from Christchurch, a distance of 485 kilometres one way. Assume that petrol will cost $1.80 per litre and that the car uses one litre every 15 kilometres; that is, 15 kilometres per litre.

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2. Compare the prices of the two special deals with the prices of individual items and show which special deal has the biggest saving.

Show all your working clearly.

Gianni’s Takeaway Pizzas

Pizzas Large pizzas $10.90 Fries $4.00 Garlic bread $4.00 Drinks (1.5 L bottle) $4.00 Desserts $3.00

Pizzas Gianni’s Special Includes: 2 large pizzas garlic bread 1.5 L soft drink

Pizzas $24.90

Mama’s Special Includes: 2 large pizzas fries 2 desserts

$25.50

3. Here is a recipe for chocolate brownies. The cost of each ingredient is also given. Remember that 1 kg = 1 000 g.

Find the cost of each item used in the recipe and the total cost of making the chocolate brownies.

Chocolate brownies

125 g chocolate 250 g chocolate costs $3.60

125 g butter 500 g butter costs $4.00

4 eggs 1 dozen eggs (12) costs $3.65

250 g sugar 1.5 kg sugar costs $3.30

100 g flour

1.5 kg of flour costs $3.20

Check your answers. Have you self-marked and corrected all of the exercises you have done so far? Were you able to answer the questions about decimals? Were you able to solve the problems with working out costs? If you had difficulty with any of these questions, phone your teacher as soon as possible.

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understanding fractions learning outcome Solve problems using fractions.

learning intentions In this lesson you will learn to:

• understand the meaning of fractions

• compare the size of fractions

• change a fraction to a decimal.

introduction Fractions are used to describe a part of something. It could be part of a whole, 1 1 such as of a pizza, or part of a group, such as of the team. 3

2

Fractions are used often in everyday life.

1 • The goal shoot in netball can play in of the court. 3 3 • The recipe for biscuits uses of a cup of sugar. 4 1

• The pattern for a dress requires 3 m of fabric. 4

naming fractions A fraction is made up of 2 numbers. The bottom number names the fraction. It tells you how many equal parts there are in the whole. FRACTION

NAME

1 2

One half

1 3

One third

1 4

One quarter

1 5

One fifth

1 10

One tenth

1 100

One hundredth

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DIAGRAM

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understanding fractions

The top number of a fraction tells you how many parts. FRACTION

2A

NAME

2 3

Two thirds

3 4

Three quarters

4 5

Four fifths

3 10

Three tenths

5 6

Five sixths

DIAGRAM

Shade in part of the diagram to show the fraction. 2 1. Colour in of the squares. 3

5 2. Colour in of the triangles. 8

7 3. Colour in of the octagons. 10

3 4. Colour in of the triangles. 4

Check your answers.

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fractions on a number line A fraction can also be shown as a position on a line. 1 The fraction is one quarter of the way between 0 and 1 on a number line. 4

1 4

0

1

2

The fraction 4 is two thirds of the way between 4 and 5 on a number line. 3 2

4

Write down the fraction shown by the arrow on each of these number lines.

4.

0

0

0

1

1

1

2

•

3.

1

•

2.

0

•

1.

•

2B

5

43

Circle the largest fraction in each group given below. Think about where the fractions would be on a number line to help you find the largest fraction. 5. 3 7

1 7

5 7

6.

1 4

1 2

1 9

7.

2 3

3 4

3 8

sevenths

0

1 ninths quarters

0

1 quarters eighths thirds

0 1

8. 2

3 8

1 eighths

1 6

sixths

0

1

Check your anwsers.

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comparing fractions

Imagine I offered you this choice:

3 'You can have of a block of chocolate,

5

13 or of 25

Which fraction would you choose?

a block of chocolate.'

You need a way of comparing these two fractions. One way is to rename both fractions so that they have the same bottom number. In this example, you could rename the fractions so that they are both out of 100. That is also useful if you want to rewrite the fractions as decimals. 3 Look at the fraction . 5

3

60

( ) 3 × 20

is the same as which is the same as 0. 6. 5 5 × 20 100

Multiplying the top and bottom lines by 20 does not change the 20 value of the fraction: = 1. 20

13 What about the fraction ? 25 52

( ) 4 × 13

13 is the same as , which is the same as 0. 52. 4 × 25 100 25

52

3 13 60 So, it is now easy to see that is bigger than , because is bigger than . 100 5 25 100

Comparing the decimals, 0. 6 is bigger than 0. 52. To write a fraction as a decimal, you can rename the fraction so that it is out of 100. Example 3 3 15 To write as a decimal: 20 = = 0.15 20 100

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understanding fractions

2C

Write these fractions as a fraction out of 100 and then as a decimal. 1 1. = = 2

100

3 2. = = 100 10

4 3. = = 5

100

9 4. = = 20

100

11 5. = = 100 25

7 6. = = 50

100

1

7. Which of these fractions is not the same as ? Change the fractions to decimals first. 5 2 10

5 20

20 100

5 25

1 8. Which of these fractions is not the same as ? 3

2 6

33 99

30 90

33 100

Check your answers.

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using a calculator to change a fraction to a decimal 3 Could you write as a decimal? 8

You can use a calculator to write a fraction as a decimal. The line in the fraction works like a division sign ÷. The top number is divided by the bottom number. 3 means 3 ÷ 4, so using your calculator, 3 ÷ 4 = 0.75. 4 17

Try this with . Using your calculator, 17 ÷ 20 = 0.85. 20 Using a calculator to change fractions to decimals is very useful for fractions that are difficult to rename as a fraction out of 100. Examples 2

(using a calculator 2 ÷ 3) = 0.6666666 … 3

= 0.67 (rounded to 2 d.p.)

5 (using a calculator 5 ÷ 6) = 0.83333333 … 6

20

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= 0.83 (rounded to 2 d.p.)

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understanding fractions

2D

Use your calculator to change these fractions to decimals. Round to 2 decimal places if rounding is necessary. 1. 3 = 25

2. 19 = 20

3. 1 = 9

4. 9 = 16

5. 1 = 12 2

6. 11 = For each group of fractions below:

• change the fractions to decimals • find the biggest fraction.

13 7. 2 = 6 = = The biggest fraction is

.

1 = = 3 8. 3 = The biggest fraction is

.

4 9 9. 7 = = = The biggest fraction is

.

3

8

8

7

3

5

15

10

11

Check your answers.

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FOR INTEREST Patterns with repeating decimals Some families of fractions form patterns when they are written as decimals. Have a look at the ninths.

1 9

= 0.11111 …

2 9

= 0.22222 …

3 9

= 0.33333 … 4

5

6

7

8

Find the decimals equivalent for . 9 9 9 9 9 Does the pattern continue? The family of elevenths also has a pattern. 1

2

3

Find the decimals for … and so on. 11 11 11 Do you see the pattern? Other families of fractions form patterns that are not so obvious. Look at the ‘sevenths’ and see if you can find a pattern.

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solving problems with fractions learning outcome Solve problems using fractions.

learning intentions In this lesson you will learn to: • find a fraction of a given amount • solve problems that involve fractions.

introduction Fractions are used in many situations. 1

• The basketball team play of their 34 games away from home. 2

• Your brother ate of the pizza. 8

• You are saving of the $180 you earned from your part-time job. 3

4 • Of the 60 people at the movie, were teenagers. 5

5

2

Let us look at some examples.

finding a fraction of an amount Example 1 A bakery makes 300 loaves of bread every day. 1 Of the loaves, are French sticks. 5

How many French sticks do they bake? 1 means 1 part out of 5. 5

Divide the 300 into 5 parts. Each part is 60. So, 1 part out of 5 is 60. 60

60

60

60

1 1 of 300 = × 300

=

= 60

5

'OF' MEANS 'TO MULIPLY'.

60

5 300 5

The bakery makes 60 French sticks.

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Example 2

3

Tanya had saved $488. She spent of her savings on a holiday. How much did she spend? 4 34 means 3 parts out of 4. Divide the $488 into 4 parts. Each part is $122. 122

122

122

3

122

3

4 of 488 = 4 × 488

= $366

Tanya spent $366 on her holiday. Example 3 The netball team scored 45 goals in their game. 4 Marama scored of the team’s goals. 9 How many goals did she score? Divide 45 into 9 parts. Each part is 5. 5

5

5

5

5

5

5

5

5

4

4

9 of 45 = 9 × 45 = 20

Marama scored 20 goals.

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solving problems with fractions

3A

Fill in the blank spaces in these problems. 1

1. There are 126 apple trees in Bob’s orchard. In a hail storm, of the trees were damaged. 3 How many trees were damaged?

Divide 126 into 3 parts. Each part is . 1

1

of 126 = × 126 = 3 3

trees were damaged. 2

2. There are 720 pupils at Ferndale High School. Of the students, travel to school by bus. 5 How many students travel by bus?

Divide 720 into 5 parts. Each part is .

2 parts equals . 2

2

of 720 = × 720 = 5 5

students travel by bus. 7

3. There are 5 270 supporters watching a rugby game. Of the supporters, are fans of the 10 Crusaders. How many of the supporters are fans of the Crusaders? Divide 5 270 into 10 parts. Each part is .

7 parts equals . 7

7

of 5 270 = × 5 270 = 10 10

supporters are fans of the Crusaders.

4. Greg and his mates go for a 48 kilometre bike ride from Nelson to Motueka. They stop for a 3 break of the way along the ride. How far from Nelson were they when they stopped 4 for a break?

Divide 48 into 4 parts. Each part is .

3 parts equals .

of 48 = × 48 = 4 4

Greg and his mates stopped for a break after kilometres.

3

3

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5. Emma has 5.2 metres of fabric. She uses of the fabric to make a shirt. 8

What length of the fabric does she use?

Divide 5.2 into 8 parts. Each part is .

5 parts equals . 5

5

of 5.2 = × 5.2 = 8 8

Emma uses

metres of fabric.

Check your answers.

using a calculator to find fractions of an amount Example 1 2 Nick has 126 shots at goal at basketball training. He is successful with of his shots. 2 A fraction like means 2 parts out of 3. It can also 3 2 On a calculator, of 126 is: 2 ÷ 3 × 126 = 84. 3

3

be read as 2 ÷ 3.

So Nick has 84 successful shots. Example 2 3 A triathlon has a total distance of 30 kilometres. The running section is of the total distance. 8

How far is the run? On a calculator, 3 ÷ 8 × 30 = 11.25 kilometres. The run is 11.25 kilometres.

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3B

Use a calculator to solve these problems. Write down the calculation you will do on your calculator (just like the examples we've done). 1

1. Kiri has $24.60 in her wallet. She spent of her money on lunch. How much did she spend? 6 1

2. A piece of timber is 3.2 metres long. Nick cuts off a piece of the total length. How long is the 5 piece of timber Nick cut off?

2 3. There is 1.8 kilograms of fish in the fridge. Sanjay uses of the fish to cook dinner. 3 How much fish does he use?

4

4. Tony and Tim play 15 games of noughts and crosses. Tim wins of the games. How many 5 games does Tim win?

5. Chris and Jane are driving 455 kilometres to Christchurch. They share the driving and Jane 5 drives of the way. What distance does Jane drive? 8

2 6. On a particular television channel, the adverts play for of every hour. How many minutes of 9 each hour do the adverts play?

3 7. Jason is baking a cake. He uses of a 500 gram block of butter. How much butter does he use? 8

2 8. Simone drinks of a 600 millilitre bottle of juice. How much juice is left? 3

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solving problems with fractions 3

9. A silver medal at the Olympic games weighs 568 grams. Only of the medal is actually 40 silver and the rest is copper. Find the weight of copper in the medal.

2

10. Kate eats of a block of chocolate. The chocolate has 30 squares. How many squares 5 of chocolate are left?

Check the answer guide.

solving problems using fractions Here are some examples of fraction problems that involve more than one step. Example 1 James orders a meat lovers pizza and his sister Claire orders a Hawaiian pizza. 7 3 James eats of his pizza and Claire eats of her pizza. 8

4

How much pizza is left over altogether? 1

1 of James’s pizza is left and of Claire’s pizza is left. 4

8

1

1

3

Adding these together → + = 8 4 8

3 There is of a pizza left over in total. 8

a

b c

Fractions on the calculator: Using the button

1 1 To find + on the calculator: 1 8 a bc + 1 4 = a bc 8

4

b

b

3 To type 2 on a calculator: 2 3 5 ac ac 5

You can use the key to change a fraction to a decimal. ab

3 a c

3 = 0.375

28

c

3 To write as a decimal: 8

b

b

8 = and press the key again. ac

8

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solving problems with fractions

Example 2 The mathematics department orders 50 muffins for morning tea. 2 3 Of the muffins, are banana, are chocolate and the rest are blueberry. 5

10

How many blueberry muffins are ordered?

2

3

7

5 + = 10 10

3

There must be 10 left. 3 So of the muffins are blueberry. 10

3

3 of 50 = × 50 10 10

= 15 15 blueberry muffins are ordered. Example 3

2 In the fridge, there is a 2.25 litre bottle of soft drink that is full. 3 1 Tony poured of the drink left in the bottle into a glass. 6

How much soft drink is now left in the bottle? 2.25 litres = 2 250 millilitres

2

2

3 of 2 250 = × 2 250 = 1 500 3

There is 1 500 millilitres of soft drink in the bottle in the fridge. 1 Tony pours of this into a glass. 6

1

1

6 of 1 500 = × 1 500 = 250 6

Tony pours 250 millilitres. The amount of soft drink now left in the bottle is 1500 – 250 = 1250 millilitre.

3C

Solve these problems. 2

1. A football team played 15 games during the season. They won of their games 3 1 and drew of their games. How many games did they lose? 5

1

3

2. I spend of my wages on rent and on food each week. What fraction of my wages is 3 10

left over?

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solving problems with fractions 1

1

1

3. In a packet of 24 smarties, are red, are yellow, are purple and the rest are green. 3 4 8

How many green smarties are in the packet?

4. A 500 millilitre drink is made by mixing orange juice, lime juice and ginger ale.

1

1

of the drink is orange juice and of the drink is lime juice. Find the amount of ginger ale 5 10 that is in the drink.

7 3 5. A 1 litre bottle of milk is full. At breakfast, of the milk is used. How much milk is left? 10 7

1

6. Sharee wants to buy a new pair of jeans that cost $78.00. The store has reduced the price by 5

2

and Shareeâ€™s mum is paying for of the reduced price as a birthday present. How much 3 does Sharee need to pay?

2 1 7. Alison has 300 trees in her orchard. Of the trees, are lemon trees and are orange trees. 3 5

The rest are lime trees. How many lime trees does she have in her orchard?

8. On the drive from Nelson to Dunedin, Mum, Dad and Sarah share the driving. Mum drove

2

1 for of the way and Dad drove of the way. What fraction of the way did Sarah drive and 7

who drove the greatest distance?

3

Check your answers.

FOR INTEREST If you have access to the internet, there are websites where you can learn about fractions and practise your skills by playing games.

You can access these sites through the online learning system.

Phone your teacher if you do not know how to access the online system.

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4

percentages learning outcome Solve problems using percentages.

learning intentions In this lesson you will learn: • understand the meaning of percentages • find percentages without and with a calculator • write a fraction as a percentage.

introduction Percentages are used in many situations. A chocolate bar has 20% extra free. The clothing shop advertises 25% off all prices in a sale. The interest that you pay on money that is borrowed from the bank might be 8.5%. If you have money saved with the bank, your money earns a percentage payment. The goods and services tax (GST) is a percentage paid to the government as tax on all goods and services sold. A percentage is a part of, or a fraction of, 100. This is useful, because it converts everything to a scale of 100, which makes it easier to make comparisons and easier to have standard rules for things like tax, discounts and payments.

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percentages

understanding percentages In the diagrams below, there are grids with 10 Ă— 10 = 100 squares. 1

Each square is = 1% = 0.01 of the total. 100 10% means 10 out of 100.

10

= = 0.1 10

35

= 0.35

7

= 0.07

10% = 100

35% = 100

7% = 100

1

A double number line is useful for seeing the link between percentages and fractions.

0

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0

1 10

2 10

3 10

4 10

5 10

6 10

7 10

8 10

9 10

1

0

20%

40%

60%

80%

100%

0

1 5

2 5

3 5

4 5

1

0

25%

50%

75%

100%

0

1 4

2 4

3 4

1

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percentage % fraction

percentage % fraction

percentage % fraction

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percentages

4A

1. Fill in the gaps in the table below. percentage

fraction

25% 50%

decimal

0.25 50 100

= 2

1

20 100

= 5

75% 20%

1

40% 60% 80% 3% 30%

0.03 30 100

3

= 10

Check your answers.

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percentages

working out easy percentages 10

1

10% is the same as = . 100 10 1

To find 10% of a number, find of the number. This is the same as dividing the number by 10. 10

10% of 180 = 180 ÷ 10 = 18

10% of 54 = 54 ÷ 10 = 5.4

THE QUICK RULE: TO FIND 10%, DIVIDE BY 10.

10% of 3 870 = 3 870 ÷ 10 = 387

Can you see how you could use this quick rule to find 20% or 5% or 90%?

Quick rules for finding percentages

To find 10%, divide by 10.

To find 20%, find 10% and double the answer 2 × 10%.

1 To find 5%, find 10% and halve the answer × 10%. 2

To find 90%, find 10% and take it away from the original amount 100% – 10%.

Let’s look at some examples of how this works. Find 20% of 280 10% of 280 = 28 20% of 280 = 56 (find 10% and then double the answer) Find 5% of 90 10% of 90 = 9 5% of 90 = 4.5 (find 10% and halve the answer) Find 90% of 250 10% of 250 = 25 90% of 250 = 250 – 25 = 225 (find 10% and subtract it from the original)

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percentages

4B

Find these percentages without using a calculator. Work out 10% first. 1. Find 10% of 360.

2. Find 10% of 56 100.

3. Find 10% of 78.

4. Find 20% of 900.

5. Find 20% of 180.

6. Find 5% of 40.

7. Find 5% of 3 600.

8. Find 90% of 300.

9. Find 90% of 80.

10. Find 90% of 84. Check your answers.

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percentages

finding a percentage of an amount Many percentage calculations are difficult to work out quickly. For these problems, use a calculator. Example 1 In a sale, the prices are reduced by 35%. A pair of jeans that normally costs $84.00 is reduced. How much will they cost in the sale? We need to find 35% of 84 Remember that ‘of’ means multiply

35

35% of 84 = 100 × 84 = 35 ÷ 100 × 84

= 29.4

The discount is $29.40. The discounted price is now $84.00 – $29.40 = $54.60 For money problems, write the answer to 2 decimal places Example 2 Money in a savings account earns interest of 6.4% per year. Jenny has $2 340 in her savings account. How much interest does she earn after one year? Interest after one year:

6.4

6.4% of 2 340 = × 2 340 100 = 6.4 ÷ 100 × 2 340

= 149.76

The money in the savings account has earned $149.76 interest.

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percentages

Example 3 Hayden would like to buy a season pass for the ski season. If he pays early, the ticket price is reduced by 15%. The price of a season pass is normally $635. How much is the reduced price?

15

Discount = 15% of 635 = × 635 100 = 15 ÷ 100 × 635

= 95.25

The price is reduced by $95.25. The new price is $635 – $95.25 = $539.75.

4C

You will need a calculator to find these percentages. Show your working. 1. Find 10% of 34 560.

2. Find 3% of 80.

3. Find 67% of 450 000.

4. Find 12.5% of 88.

5. Find 15% of 180.

Check your answers.

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percentages

writing fractions as percentages Example 1 In her netball game, Irene had 46 attempts at goal and was successful with 37.

She likes to compare her shooting rate in all her games and it is easiest to do this using a percentage.

37

37 out of 46 can be written as a fraction . 46 To change this to a percentage, multiply by 100. Remember that a percentage is a fraction of 100.

37

46 × 100 = 80.4 %

Irene had a shooting success rate of 80.4% in her game. Use your calculator for this example: 37 ÷ 46 × 100 = b

Or, if you use the fraction key a c on the calculator: b

b

ac a c 37 46 × 100 =

(to change to a decimal answer)

Example 2 On a warm summer day, an ice cream shop sold 23 hokey pokey ice creams, 27 vanilla, 12 boysenberry and 21 chocolate. What percentage of the ice creams sold were vanilla? The total number of ice creams sold = 23 + 27 + 12 + 21 = 83. 27 of these were vanilla. 27

The fraction of vanilla ice creams = . 83 As a percentage:

To change a fraction to a percentage, multiply by 100. 27 83

× 100 = 32.5% (rounded to 1 d.p.)

(On the calculator 27 ÷ 83 × 100 = 32.5% )

32.5% of the ice creams sold were vanilla.

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percentages

4D

You will need a calculator. If you need to round the answers, round to 2 decimal places. 1. Change these fractions to percentages. 3 8 (3 ÷ 8 × 100)

37.5 %

3 25 4 5 19 20 28 31 145 283 356 512 1 345 5 000

For the examples below, show your working. 2. A pie shop sold 58 steak pies out of a total of 145 pies sold. What percentage of the pies sold were steak pies?

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percentages

3. A car sales yard requires a deposit of $3 250 on a car for sale at $9 750. What percentage is the deposit required?

4. In a class of 30 students, 4 students are left-handed. What percentage of students are left-handed?

5. Sam threw a die (one dice) 50 times and recorded the numbers that turned up. He threw 8 sixes in total. What percentage of the dice throws were NOT sixes?

6. In a netball game, Miriama shot 18 goals and Kim shot 13 goals. What percentage of the goals did Kim shoot?

7. In a probability experiment, Joshua tossed a coin many times. The coin landed on heads 43 times and tails 37 times. What percentage were heads?

8. In an election for a Member of Parliament, the votes in one electorate were: National 5 781 Labour 6 788 Greens 543 MÄ ori party 1 067 All other votes 863

What percentage of the votes did the MÄ ori party receive?

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5

discounts, payments and gst learning outcome Solve problems using percentages.

learning intentions In this lesson you will learn to: • increase or decrease a number by a percentage • calculate GST.

introduction A percentage discount is an amount that is taken off the usual price. In a sale, prices are reduced by a percentage. A student discount is a percentage taken off the usual price. Sometimes there is a reduction in price for paying on time. Savings in a bank account earn a percentage in interest that is added to the amount of money (principal) in the account. For money that is borrowed, a percentage is paid as the cost of borrowing the money. GST is a goods and services tax that is added to the price of things we buy. GST has recently increased from 12.5% to 15% (October 2010). This tax is paid to the government by the shopkeeper or the person who provides the service.

increase or decrease by a percentage An example involving interest Nick puts $500 in his bank account, which earns 6% interest per year. How much money will he have in his account at the end of the year?

6 6% of 500 = 100 × 500

= 6 ÷ 100 × 500

= 30

Nick has earned $30 interest on his savings. Nick’s total savings are now $530.

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discounts, payments and gst

A GST example In October 2010, GST increased from 12.5% to 15%. Retailers now need to add 15% to their prices, rather than 12.5%. This money is paid to the government. A packet of biscuits costs $1.80 before GST is added:

15% of $1.80

15 = × 1.80 100

= 15 ÷ 100 × 1.80

= $0.27

27 c is added to the price of the packet of biscuits. The price including GST = 1.80 + 0.27 = $2.07.

gst the quick way Can you think of a quick way of adding GST without a calculator? Try 10% plus 5%. Thinking of $1.80:

10% of $1.80

= 0.18

5% of $1.80

= 0.09

15% of $1.80

= $0.18 + $0.09 = $0.27

Try this with $600:

10% of $600

= $60

5% of $600

= $30

15% of $600

= $60 + $30 = $90

An example of a percentage discount In a sale, the price of a pair of jeans is reduced by 30%. They normally cost $86. How much are they reduced by in the sale? 30% of 86

30 = 100 × 86

= 30 ÷ 100 × 86

= 25.8

The price is reduced by $25.80. The sale price is $60.20.

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Remember to write answers with money to 2 decimal places.

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discounts, payments and gst

5A

1. A 300 gram block of chocolate has 10% extra added. Find the extra amount added and the new weight.

2. A hairdresser charges $60 for a cut and colour before 15% GST is added. How much GST is added?

3. A plumber works on a new house and sends an account. He has added 15% GST to the total cost of $2 600. Add 15% GST to $2 600.

4. The local music shop gives a student discount of 20%. How much is the student discount on a CD that normally sells for $22?

5. A car dealer buys a car for $3 800 and sells it for 35% more. What is the selling price of the car?

6. In a sale, the price of a DVD that normally sells for $25 is reduced by 30%. What is the new price for the DVD?

7. After training for a few weeks, a runner reduces the time he takes to run 5 kilometres by 16%. Before training he ran the 5 kilometres in 25 minutes. How long did it take to run 5 kilometres after training?

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discounts, payments and gst

8. A 280 millilitre bottle of shampoo has 10% extra added for free. What is the volume with the extra added?

9. Find the interest paid on savings of $2 350 at an interest rate of 5.7% per year. a. After one year.

b. After 2 years, assuming that the interest is added to the savings after the first year.

10. GST increased from 12.5% to 15% in October 2010. a. Increase $180 by 12.5%.

b. Increase $180 by 15%.

c. How much extra was added to $180 when GST increased from 12.5% to 15%?

Check your answers.

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6

percentage increases and decreases learning outcome Solve problems using percentages.

learning intentions In this lesson you will learn to: • calculate an increase of a given amount as a percentage • calculate a decrease of a given amount as a percentage.

introduction From time to time, many things increase. The population of countries, cities and towns; the production of dairy products in New Zealand; the number of apples on a tree; the size of the crowd at a football game; and so on. Other things decrease. The time it takes for elite athletes to run 100 metres or the value of the car you bought last year. For a fair comparison between given amounts, these increases and decreases are written as percentages. That way, the scale is the same. It makes it easier and fairer to make a comparison.

increases and decreases written as percentages Think of an increase, such as an increase in the population of a country or an increase in the price of milk, that you want to write as a percentage increase. In the same way, think about a decrease, such as a fall in the number of houses sold per month by a real estate agent or a reduction in the number of road accidents, that you want to write as a percentage decrease. In these examples, the increase or decrease is compared to the original amount and then changed to a percentage.

writing an increase as a percentage In 2010, New Zealand had a population of 4.37 million. The population of New Zealand is predicted to increase to 5.75 million by 2061.

What is the expected percentage increase in the population during this time?

Increase is 5.75 – 4.37 = 1.38. The population increases by 1.38 million. The original population is 4.37 million.

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percentage increases and decreases

increase 1.38 million = original 4.37 million

To change this to a percentage, multiply by 100.

1.38 × 100 = 31.6 % ( 1 d.p.) 4.37

1.38 ÷ 4.37 × 100 = 31.6%

The population is expected to increase by 31.6% between 2010 and 2061.

writing a decrease as a percentage For a special promotion, the price of a flight from Nelson to Wellington is reduced from $109 to $69. What is the percentage decrease? The price decrease is $40. The original price is $109.

decrease 40 original = 109

To change this to a percentage, multiply by 100.

40 109 × 100 = 37% ( rounded to 2 s.f.)

40 ÷ 109 × 100 = 37%

The price of the flight has been reduced by 37%. 6A

You will need a calculator. Find these percentage increases and decreases. Show your working. 1. Find the percentage increase if 80 is increased to 84.

4 increase × 100 = × 100 80 original

=

=

2. Find the percentage decrease from 180 to 130.

46

50 decrease × 100 = × 100 180 original

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percentage increases and decreases

3. A music shop sold 245 CDs last week and 313 CDs this week. What is the percentage increase in sales?

4. Last year, Zoe picked 13 kilograms of feijoas from her tree and this year she picked 17 kilograms. What is the percentage increase in the weight of feijoas Zoe picked?

5. Joe is on a diet. Last month he weighed 124 kilograms and this month he weighs 119 kilograms. What is the percentage decrease in his weight?

6. Ashleigh earns $15.30 per hour and her pay is going to go up to $16.83 per hour. What is the percentage increase in her pay?

7. Jake bought a car for $5 200 and sold it for $3 800. What is the percentage decrease in the value of the car?

8. Stephen is training for a triathlon. This week he swam 30 lengths of the pool every day. Last week he swam 25 lengths of the pool every day. What is the percentage increase in the number of lengths Stephen swam each day?

Check your answers.

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review activity

7

learning outcome

Review your understanding of decimals, fractions and percentages.

learning intentions In this lesson you will learn to: • revise your work on decimals, fractions and percentages • answer questions for your teacher to assess.

introduction In these problems, make sure that you show all the steps in your working. Some of these problems are long and there is a lot of reading. Read the question more than once before you start answering the question. If the question is long, write down the separate parts you need to do to answer the question.

checkpoint 7A

1. Simon is building a small deck at home. He needs to buy different types of timber for the piles, bearers, joists and decking.

Work out the total cost of the timber he needs to buy. timber

amount he needs

cost

total cost

Piles

12

$13.40 each

Bearers

3 lengths of 3.6 m

$17.95 per length

Joists

4 lengths of 3.6 m

$13.95 per length

Decking

70 m

$2.25 per m total cost

2. Anna wants to buy a new cellphone that normally sells for $369. • The price is reduced by 20% for a special deal.

48

5 • She has saved of the reduced price. 8

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review activity

• She will borrow the rest from her dad. How much money does Anna need to borrow from her dad?

3. A triathlon has three sections: swimming, running and riding a bike. The total distance is 40 kilometres. • The swimming section is 2% of 40 kilometres.

1

• The run is of 40 kilometres. 5

How far, in kilometres, is the bike ride?

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review activity

4.

Sione is planning to go on a holiday. The package deal costs $1 640 in total. The travel agency requires a 30% deposit, which he is paying for straight away. He is going to save up to pay for the rest. He plans to save $82 per week. How many weeks will it take Sione to save for the rest of the money he needs? Remember to show all the steps in your working.

5. Armani has a shoe shop. He buys in 60 pairs of shoes in a certain style.

• Armani pays $54 (cost price) for each pair of shoes.

• He sells of the shoes at $92 per pair. 3

2

• The rest of the shoes he sells at 25% reduced price in a sale.

Work out how much profit Armani makes by selling these shoes. Write this profit as a percentage of the amount Armani initially spent on buying the shoes.

Remember to show all the steps in your working.

Check your answers.

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review activity

assessment task practice Your teacher will mark this work. This assessment does not count for achievement standard credits. In this booklet, you have been working towards Achievement Standard 91026 ‘Apply numeric reasoning when solving problems’, which will be assessed after you have completed MX1012 and MX1013. Show all your working. Write your working and answers in the spaces provided. You will need a calculator. 7B

1. Dressmaking Gemma wants to work out the cost of making a jacket. Here is the list of things she needs to buy.

• • • •

3.2 metres of fabric normally priced at $12.50 per metre. The price of the fabric is reduced by 20%. 0.6 metres of interfacing at $3.50 per metre. 6 buttons at 75 c each. A reel of thread at $3.10.

Fabric Interfacing Buttons Thread total cost

Find the total cost of the jacket.

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review activity

2. Jake's motorbike

Jake wants to buy a motorbike that is for sale at $4 200.

He will pay a deposit now and will borrow the rest at 8% interest per year. 3

1

He wants to pay off the loan in 1 year (52 weeks).

He has budgeted $60 per week to pay off the money he has borrowed.

Will that be enough to pay off the loan in 1 year? 1

deposit 3

Amount he needs to borrow

Interest cost for borrowing

Total amount he needs to repay

$60 a week for 52 weeks

Is $60 per week enough to pay off the bike in 1 year?

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review activity

3. Finding the best deal

Jayden would like to buy the latest X-box, so he looks around for the best deal. Calculate the cost of each deal and find the cheapest option for Jayden.

Deal 1: $250 deposit and 52 weekly payments of $7.80.

Deal 2: $789, less 25% discount if you pay cash.

Deal 3: Full price of $720. Pay now and 12 equal monthly payments of $50. 3

Which is the cheapest deal?

1

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review activity

4. Christmas shopping

The Thompson twins, Nicky and Paris, have saved $120 to buy Christmas presents for their family.

They need to buy for Mum, Dad, Grandma and their brother Matt. The money they have left over will be split between them to buy presents for each other.

• They spend of the money on a present for Mum. 5

1

• 18% of the money goes on a present for Dad.

• 0.15 of their money goes on a present for Grandma.

• After buying these 3 presents, they work out how much they have left.

• Then they spend of the money they have left on a present for their brother Matt. 8

How much money do they have left? Set your work out clearly, showing what is being calculated at each step and explaining any decisions you make.

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review activity

5. Trip to Nelson

Harry is driving from Christchurch to Nelson – a distance of 420 kilometres.

4 • He stops for petrol in Kaikoura, which is of the distance from Christchurch.

• The petrol tank holds 55 litres and

• Harry buys 40 litres of petrol at Kaikoura.

• He drives the rest of the way to Nelson and uses 1 litre of petrol every 12 kilometres.

How much petrol is left in the tank when Harry gets to Nelson?

Set your work out clearly, showing what is being calculated at each step and explaining any decisions you make.

Please return your book to be assessed.

7 2 is full 11

at Kaikoura before he buys more petrol.

Your teacher will assess this work.

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8

answer guide 1. 1A

decimals

1.

3.7

2.

63.7

3.

1.15

4.

0.71

5.

0.9

6.

3.2

7.

Any number from 3.71 to 3.79; for example, 3.75 (to 2 d.p.).

8. 1.32

1.3

1.29

1.26 1.19 1.03

9. a. Putting the times in order with fastest first:

58.205

58.962 59.003

59.468 60.063 61.004 61.096

Third fastest is then 59.003. b. Three one hundredths of a second = 3/100 sec = 0.03 sec

1B

1.

6.6 (1 d.p.)

2.

3.4 (1 d.p.)

3.

0.1 (1 d.p.)

4.

1.1 (1 d.p.)

5.

17.0 (1 d.p.)

6.

2.34 (2 d.p.)

7.

20.33 (2 d.p.)

8.

12.18 (2 d.p.)

58.205 â€“ 0.03 = 58.175 sec

solving word problems (page 7)

Multiply

Subtract

Add

Subtract

Divide

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answer guide

1C

1D

1.

Subtract: 2.83 – 1.06 = 1.77, so 1.77 m

2.

Add: 5.45 + 1.35 = 6.80, so $6.80

3.

Divide: 5.6 ÷ 8 = 0.7, so 0.7 kg

4.

Add: 35.7 + 6.55 = 42.25 kg

5.

Add: 6.82 + 1.6 = 8.42, so 8.42 km

6.

Subtract: 126 350 – 26 500 = 99 850, so 99 850 people

7.

Multiply: 35.40 × 8 = 283.20, so $283.20

8.

Subtract: 1.85 – 1.62 = 0.23, so 0.23 m

9.

Multiply, add, subtract: 2 × 1.40 + 0.7 = $3.50 5.00 – 3.50 = $1.50

10.

Divide: = 3 = 0.67 pizzas 12

1.

Cost of petrol Distance = 485 × 2 = 970 km Petrol usage = 970 ÷ 15 = 64.66 L Cost = 64.66 × 1.80 = $116.40

Cost of cabin One night = 60 + 4 ×15 = $120 Six nights = 120 × 6 = $720

Total cost = 116.40 + 720 + 480 = $1 316.41

Each person cost = 1316.41 ÷ 6 = $219.40 (2 d.p.)

8

2

2. Gianni’s Special Cost of 2 large pizzas + garlic bread + 1.5 L drink = 2 × 10.90 + 4.00 + 4.00 = $29.80 Difference = 29.80 – 24.90 = $4.90 saving Mama’s Special Cost of 2 large pizzas + garlic bread + 1.5 L drink = 2 × 10.90 + 4.00 + 6.00 = $31.80 Difference = 31.80 – 25.50 = $6.30 saving

Mama’s Special has the biggest saving.

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answer guide

3.

Cost of: 125 g chocolate 125 g butter 4 eggs 250 g sugar 100 g flour

Total cost

2. 2A

2B

= 3.60 ÷ 2 = $1.80 = 4.00 ÷ 4 = $1.00 = 3.65 ÷ 3 = $1.22 = 3.30 ÷ 6 = $0.55 = 3.20 ÷ 15 = $0.21 = $4.78

understanding fractions

1.

Shade any 6 squares.

2.

Shade any 5 triangles.

3.

Shade any 7 octagons.

4.

Shade any 9 triangles.

1.

3 10

2.

8 or 4 10 5

3.

7 10

4.

12

5.

5 7

6.

1 2

7.

3 4

8.

1 2

58

1

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answer guide

2C

50

1.

1 2

= 100

2.

3 10

= 100

= 0.3

3.

4 5

= 100

80

= 0.8

4.

9 20

= 100

45

= 0.45

5.

11 25

= 100

= 0.44

6.

7 50

= 100

14

= 0.14

7.

5 20

=

30

44

= 0.5

1 4

33

8. 100

2D

1.

3 25

= 0.12

2.

19 20

= 0.95

3.

1 9

= 0.11

4.

9 16

= 0.56

1

5. 12 = 0.08 2

6. 11 = 0.18 2

6

7. 3 = 0.67 7 = 0.86

13 15

= 0.87

13

The biggest fraction is . 15

Decimals have been rounded to 2 decimal places. 3

1

8. 8 = 0.38 3 = 0.33

3 10

= 0.30

Decimals have been rounded to 2 decimal places. 7

4

3

The biggest fraction is . 8

9

7

9. 8 = 0.88 5 = 0.80 11 = 0.82 The biggest fraction is . 8 Decimals have been rounded to 2 decimal places.

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answer guide

3. 3A

solving problems with fractions

1. Each part is 42.

1

1

of 126 = × 126 = 42 3 3

42 trees were damaged. 2.

Each part is 144.

2 parts equals 288.

of 720 = × 720 = 288 5 5

2

2

288 students travel by bus. 3.

Each part is 527.

7 parts equals 3 689.

of 5270 = × 5270 = 3 689 10 10

7

7

3 689 supporters are fans of the Crusaders. 4. Each part is 12.

3 parts equals 36.

of 48 = × 48 = 36 4 4

3

Greg stopped for a break after 36 kilometres.

5.

Each part is 0.65.

5 parts equals 3.25.

of 5.2 = × 5.2 = 3.25 8 8

3B

3

5

5

Emma uses 3.25 metres of fabric.

1

1.

× 24.60 = $4.10 6

2.

5 × 3.2 = 0.64 m

1 2

3. 3 × 1.8 kg = 1.2 kg 4

4.

5 × 15 = 12 games

5.

8 × 455 = 284.38 ≈ 284 km (rounded to the nearest km)

6.

9 × 60 = 13.33 min (2 d.p.)

60

5 2

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answer guide 3

7.

8 × 500 = 187.5 g

8.

1 3 × 600 = 200 mL is left in the bottle. 3

9.

3 40

1

× 568 = 42.6 g

568 − 42.6 = 525.4 g

3C

2

10.

5 × 30 = 18 squares

1.

3 + 5 = They lost of their games. 15 15

2

2 15

1

2

13

× 15 = 2 They lost two games.

1

3

19

19

11

2.

3 + 10 = 30 I have 1 − = . 30 30

11 I have of my wages left over. 30

3.

3 + 4 + 8 = 24 The rest are green 1 − 24 = . 24

× 24 = 7 Seven out of the 24 smarties are green. 24

4.

Orange juice is × 500 = 100 mL, lime juice is × 500 = 50 mL, 10 5

1

1

1

17

17

7

7

1

1

ginger ale is 500 − 150 = 350 mL. 7

5.

7 of 10

3 × 700 = 300 mL There is 700 − 300 = 400 mL of milk left. 7

6.

15 × 78 = $15.60 Price is reduced by $15.60.

New price is 78 − 15.60 = $62.40.

Mum will pay × 62.4 = $41.60, Sharee pays 62.40 − 41.60 = $20.80. 3

7.

× 300 = 200 lemon trees 3

1 L is × 1000 = 700 mL 10

2

2

1 × 5

300 − 260 = 40. There are 40 lime trees.

8.

7 + 3 = Sarah drove of the distance. 21 21

2

2

300 = 60 orange trees

1

13

8

13

8

= 0.333 21 = 0.381 7 = 0.286 21

8 Sarah drove the greatest distance because is the biggest fraction. 21

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answer guide

4.

solving problems with fractions

percentage

4A

fraction

25%

25 100

= 4

50%

50 100

= 2

75%

75 100

= 4

20%

20 100

= 5

40%

40 100

= 5

60%

60 100

= 5

0.6

80%

80 100

= 5

4

0.8

30%

MX1011

1

0.25

1

0.5

3

1

2

3

3 100

3%

62

decimal

30 100

0.75 0.2 0.4

0.03 3

= 10

0.3

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answer guide

4B

4C

1.

360 ÷ 10 = 36

2.

56 100 ÷ 10 = 5 610

3.

78 ÷ 10 = 7.8

4.

900 ÷ 10 = 90

→ 90 × 2 = 180

→

20% of 900 = 180

5.

180 ÷ 10 = 18

→ 18 × 2 = 36

→

20% of 180 = 36

6.

40 ÷ 10 = 4

→ 4 ÷ 2 = 2

→

5% of 40 = 2

7.

3 600 ÷ 10 = 360 → 360 ÷ 2 = 180 →

5% of 3 600 = 180

8.

300 ÷ 10 = 30

→ 300 − 30 = 270 →

90% of 30 = 270

9.

80 ÷ 10 = 8

→

90% of 80 = 72

10.

84 ÷ 10 = 8.4

→ 84 − 8.4 = 75.6 →

1.

10 ÷ 100 × 34 560 = 3 456

2.

3 ÷ 100 × 80 = 2.4

3.

67 ÷ 100 × 450 000 = 301 500

4.

12.5 ÷ 100 × 88 = 11

5.

15 ÷ 100 × 180 = 27

© te aho o t e k ur a p o un a m u

80 − 8 = 72

→

90% of 84 = 75.6

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answer guide

4D

1.

3 25

3 ÷ 25 × 100 = 12%

4 5

4 ÷ 5 × 100 = 80%

19 20

19 ÷ 20 × 100 = 95%

28 31

28 ÷ 31 × 100 = 90.32%

145 283

145 ÷ 283 × 100 = 51.24%

356 512

356 ÷ 512 × 100 = 69.53%

1 345 5 000

1 345 ÷ 5 000 × 100 = 26.9%

2.

58 ÷ 145 × 100 = 40%

3.

3 250 ÷ 9 750 × 100 = 33.33% (2 d.p.) or 33 % 3

4.

4 ÷ 30 × 100 = 13.33% (2 d.p.) or 13 % 3

5.

42 were not sixes → 42 ÷ 50 × 100 = 84%

6.

13 ÷ 31 × 100 = 41.94%

7.

43 ÷ 80 × 100 = 53.75%

8.

1 067 ÷ 15 042 × 100 = 7.09%

64

1

1

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answer guide

5. 5A

discounts, payments and gst

1.

10% of 300 = 10 ÷ 100 × 300 = 30 Thirty grams is added, so the new weight is 330 grams.

2.

15% of 60 = 15 ÷ 100 × 60 = 9 $9.00 GST is added.

3.

15% of 2 600 = 15 ÷ 100 × 2 600 = 390 (GST) 2 600 + 390 = $2 990 total cost

4.

20% of 22 = 20 ÷ 100 × 22 = 4.4 $4.40 is the discount.

5.

35 ÷ 100 × 3 800 = 1 330 The selling price is $1 330 + $3 800 = $5 130.

6.

30 ÷ 100 × 25 = 7.5 The discount is $7.50, the new price is $25 – $7.50 = $17.50.

7.

16% of 25 = 16 ÷ 100 × 25 = 4 min The time is reduced by 4 minutes, the new time is 21 minutes.

8.

10 ÷ 100 × 280 = 28 mL The new volume is 280 + 28 = 308 millimetres.

9. a. 5.7 ÷ 100 × 2 350 = 133.95 $133.95 interest. The new total is $2 483.95. b. 5.7 ÷ 100 × 2 483.95 = 141.59 $141.59 is the interest in the second year. 10. a. 12.5 ÷ 100 × 180 = 22.5 → 180 + 22.5 = $202.50

b. 15 ÷ 100 × 180 = 27 → 180 + 27 = $207

c. 207 − 202.50 = $4.50

6. percentage increases and decreases 6A

1.

4 ÷ 80 × 100 = 5%

2.

50 ÷ 180 × 100 = 27.78%

3.

The increase in sales is 313 − 245 = 68. As a percentage increase 68 ÷ 245 × 100 = 27.76%.

4.

The increase is 17 − 13 = 4 kg. As a percentage increase 4 ÷ 13 × 100 = 30.77%.

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answer guide

5.

The weight loss is 124 − 119 = 5 kg. As a percentage loss 5 ÷ 124 × 100 = 4.03%.

6.

The wage rate increase is $16.83 − $15.30 = $1.53. As a percentage increase 1.53 ÷ 15.30 × 100 = 10%.

7.

The decrease in price is 5 200 − 3 800 = 1 400. As a percentage decrease 1 400 ÷ 5 200 × 100 = 26.92%.

8.

The increase in lengths is 5. As a percentage increase 5 ÷ 25 × 100 = 20%.

7. review activity 7A

1.

Piles 12 × 13.40 = $160.80 Bearers 3 × 17.95 = $53.85 Joists 4 × 13.95 = $55.80

Decking 70 × 2.25 = $157.50

Total cost of the timber = $427.95

2. The price is reduced by 20% for a special deal → 20 ÷ 100 × 369 = $73.80 $369 – $73 .80 = $295.20 5 She has saved → × 295.20 = $184.50 8

She will borrow the rest from her dad → $295.20 – $184.50 = $110.70

3.

The swimming section is 2% of the total distance → 2 ÷ 100 × 40 = 0.8 km 1

The run is of the total distance → 1 ÷ 5 × 40 = 8 km 5

The bike ride is 40 – 8.8 = 31.2 km.

4.

Deposit: 30 ÷ 100 × 1640 = $492 Left to pay: 1640 – 492 = $1148 Amount to pay ÷ amount per week → 1148 ÷ 82 = 14

It takes Sione 14 weeks to save the money.

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answer guide

5.

Armani pays $54 (cost price) for each pair of shoes.

54 × 60 = $3 240

2 He sells of the shoes at $92 per pair. 3 2

× 60 × 92 = $3 680 3

The rest of the shoes he sells at 25% reduced price in a sale. 20 pairs are left to sell. The discount is 25 ÷ 100 × 92 = $23. The reduced price is 92 − 23 = $69.

20 pairs at the reduced price → 20 × 69 = $1 380

Total sales → 3 680 + 1 380 = $5 060

Profit is → $5 060 – $3 240 = $1 820

Percentage profit is × 100 = × 100 = 56% (rounded to nearest whole number) cost of shoes

© te aho o t e k ur a p o un a m u

profit

1 820 3 240

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67

acknowledgements Every effort has been made to acknowledge and contact copyright holders. Te Aho o Te Kura Pounamu apologises for any omissions and welcomes more accurate information. Photos: Map of New Zealand – ©Mapsdata Science iStock: Cover: Numbers - 6080650 Runner - 14193439 Another runner - 5980080 Chocolate - 13385750 Fabric - 13237422 French stick bread - 13234402 Netball ball - 12624653 Pie - 12335582 Party - 12290953 Hot chocolate - 11916711 Hands holding netball - 9867893 Builder - 7314073 Pizza - 6559513 Netball in hoop - 6250469 Basketball game - 1140606 Apple - 14459377 Pizza in box - 14193439

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Solve problems involving decimals. Solve problems involving fractions. Solve problems involving percentages. Solve word problems involving decimals, fractions and/or percentages.

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