Issuu

Monique Miotto

Tracey MacBeth-Dunn

ACTIVE MATHS

Australian Curriculum edition

HomeworkProgram

Series editor and lead author

Author Contributing authors

Craig

Loretta

Ingrid

Andrew

Lisa

Monique Miotto

Tracey MacBeth-Dunn

Natalie Caruso

Dina Antoniou

Blake

Carter

Bozenna Graham

Kemp

Mark

Sinibaldi

This edition published in 2021 by

Matilda Education Australia, an imprint of Meanwhile Education Pty Ltd

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Publication data

Author: Miotto, Monique and others

Title: Active Maths 9: Homework Program (Australian Curriculum edition)

ISBN: 978 1 4202 3067 3

Publisher: Colin McNeil and Peter Saffin

Project editor: Eve Sullivan

Editor: Laura Davies

Illustrator: Paul Lennon (cartoons)

Cover designer: Dim Frangoulis

Text designer: Dim Frangoulis

Production control: Loran McDougall

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Algebra and equations 1

For 1–2, list the like terms.

[Like terms]

For 3–7, simplify the expressions by collecting the like terms.

[Simplify like terms]

[Simplify expression]

[Algebraic fractions]

[Expand: distributive law]

[Expand: distributive law] [Algebraic fractions]

[Substitution]

[Substitution: formula]

Tom’s wage, W, is represented by the formula: W = 50 + 17h where h is the number of hours he works. Find his

[Simplify expression]

For 21–30, the simple interest, I, on an amount of money, P, that is invested at a certain interest rate, r%, for a certain time, T, can be calculated using: I = PrT 100 .

21 Find the simple interest, I, if:

[Substitution: simple interest]

a P = 400, r = 5% and T = 2.

b P = 10 000, r = 4.3% and T = 6.

c P = 4500, r = 7% and T = 3.

For 23–27, find the simple interest earned.

23 $7000 is invested at 4% p.a. for 3 years.

[Substitution: simple interest]

24 $1400 is invested at 3 1 2 % p.a. for 2 years.

[Substitution: simple interest]

25 $230 000 is invested at 5.6% p.a. for 3 1 2 years.

[Substitution: simple interest]

26 $3900 is invested at 7% p.a. for 6 months.

[Substitution: simple interest]

22 Saul was asked to calculate the simple interest earned if $1600 is invested at 3% p.a. for 5 years. His working is shown here:

I = ? P = 1600 r = 3% = 0.03 n = 5

I = PrT 100 I = 1800 × 0.03 × 5 100 = 2.7

The simple interest is $2.40.

a His answer is wrong. How would he have known this?

b What did he do wrong?

c What is the correct answer?

[Substitution: simple interest]

27 $82 000 is invested at 2.1% p.a. for 15 months.

[Substitution: simple interest]

28 If $1500 is invested at 4.1% p.a. for 2 years, calculate:

[Substitution: simple interest]

a The simple interest earned.

b The total amount at the end of the 2 years.

29 Juanita is saving for a house. She invests $23 000 at 6.5% p.a. simple interest. Calculate the total amount she will have at the end of 3 years.

[Substitution: simple interest]

30 Melissa wants to go on a $12 000 package holiday in 18 months. If she invests $10 000 at 8.2% p.a. would she have enough money? Explain.

[Substitution: simple interest]

Algebra and equations 2

1 Circle the linear equation.

[Linear equation]

2 Check, using substitution, which of the following equations have x = –2 as a solution.

[Substitution: check answer]

3–18, solve for the unknown.

[Solve: 1 step]

[Solve: 2 step]

[Solve: pronumeral both sides]

[Solve: brackets]

[Solve: complex]

Solve the following equation, giving your answer as a fraction: 2 7 31 5 pp =

[Solve: complex]

[Rearrange formula] [Rearrange formula]

Solve: 5(2n + 3) = 3(4n – 1)

Rearrange the formula to make h the subject: A bh = 2

Rearrange the formula to make r the subject: A = 2prh

23 If y = 4x + 1, a write an equation to find x if y = 19.

[Substitution: solve equation]

b solve to find x

24 The formula for finding the circumference, C, of a circle is C = 2pr

[Substitution: solve equation]

a If the radius is 5 cm, find C to 1 decimal place.

b If C = 49.6, what is the radius? Answer correct to 1 decimal place.

For 25–28, the simple interest, I, on an amount of money, P, that is invested at a certain interest rate, r%, for a certain time, T, can be calculated using: I = PrT 100 .

25 For each of the following substitute the given values to get an equation and then solve it.

[Simple interest: solve equation]

a Find I if P = 200, r = 5 and T = 3.

b Find P if I = 150, r = 4 and T = 2.

26 For each of the following substitute the given values to get an equation and then solve it.

[Simple interest: solve equation]

a Find r if I = 180, P = 3000 and T = 1.5.

b Find T if I = 29 400, P = 120 000 and r = 3.5.

27 How long will Leo need to invest $1700 at 6.5% p.a. if he wishes to make $550? Answer correct to the nearest year.

[Simple interest: solve equation]

28 In 3 years Rani wants to buy into a business for $100 000. If she wants to invest her inheritance of $85 000, what simple interest rate will she need to get if she is to have enough money? Answer correct to 2 significant figures.

29 Martha’s phone plan charges her 10 cents per call plus 3 cents per second.

[Linear equation: application]

a Write an equation for the cost of a call (C) in terms of the length of the call (s) in seconds.

b How much will Martha have to pay for a 2-minute call?

c How long can she talk for $1.30?

30 Mary, a car sales person, receives a base salary of $450 per month and a commission of $100 for each car sold.

[Linear equation: application]

a Write an equation for Mary’s monthly income (I) in terms of the numbers of cars sold (n).

b How much will Mary earn in the month of December if she sells 12 cars?

c How many cars did she sell if she earned $2950?

Macmillan Active Maths 9 Investigation

Algebra connections

We use pronumerals (letter symbols) to represent an unknown quantity in our working in much the same way as you might use a box in a number puzzle. We also use pronumerals to represent quantities that can take a range of values, allowing us to describe a variety of situations with just one statement. In this investigation, we will review some of the different ways pronumerals are used.

1 Write equations to represent the following riddles, using the pronumerals suggested, and then solve the riddles.

2 Use pronumerals to complete the missing labels in the following figures. a b [Write an equation] [Interpret a diagram] x x

a Double a mystery number (n) is equal to its square. What is the number?

b The sum of two consecutive counting numbers is 19. Let the smaller number be s. What are the two numbers?

c The product of two consecutive even numbers is 224. Let the smaller number be e. What are the numbers?

d The length of a rectangle is triple its width. Let the width be w cm. The perimeter of the rectangle is 96 cm. What is the length of the rectangle?

3 For each of the following tables write a rule for y in terms of x.

[Describe a pattern]

4 Give the x- and y-coordinates of the points shown below:

[Interpret a graph]

A ( , ) B ( , )

C ( , ) D ( , )

E ( , ) F ( , )

G ( , ) H ( , ) I ( , )

5 Gustavo wants to earn $4000 interest over 5 years. He can get 4% p.a. simple interest. He has started the calculation below to work out how much he needs to invest. Complete it to find out how much he needs to invest.

[Use a formula]

He should invest $

6 The graphs below show the volumes (V) of various containers as a function of their heights (h). Match the containers to their graphs by drawing an arrow.

[Interpret relationships]

7 Consider this sequence of numbers: 1, 5, 9, 13, 17, 21, 25, ...

[Look for patterns]

[Generalise]

Numbers in a sequence are called terms.

a Find the difference between each consecutive pair of terms. What do you find?

b This type of sequence is called an arithmetic sequence. Let d represent the difference between consecutive terms. What is the value of d?

c We can represent the value of the nth term in this sequence as tn where n = 1, 2, 3, ... For example, t3 = 9. What is t5?

d There are two ways of describing a sequence. The first way is to write a rule showing how to get from one term to the next. This is called a difference equation. Complete this difference equation:

tn+1 = tn , where n = 1, 2, 3, ...

e The other way of describing a sequence involves writing a rule for the nth term. Complete the rule for the nth term of this sequence:

tn = t1 + ( – 1)4, where n = 1, 2, 3, ... and t1 is the first term of the sequence.

f Use this formula to find the 60th term of this sequence.

8 Consider this sequence of numbers: 2, 6, 18, 54, 162, ...

[Look for patterns]

[Generalise]

a Find the difference between each consecutive pair of terms. What do you find?

b Compare the ratios of each term to the one before, e.g. 6/2. What do you find?

c This type of sequence is called a geometric sequence. Let r represent the common ratio between consecutive terms. What is the value of r?

d What is t5 of this sequence?

e Complete the difference equation for this sequence:

tn+1 = tn , where n = 1, 2, 3, ...

f Complete the rule for the nth term of this sequence:

tn = t1 × ( – 1), where n = 1, 2, 3, ... and t1 is the first term of the sequence.

g Use this formula to find the 12th term of this sequence.

Macmillan Active Maths 9 Algebraic expressions

Algebraic expressions can be used to describe the patterns in geometrical shapes such as the relationship between the circumference and radius of a circle. In this task, you will write, simplify and evaluate some common algebraic expressions used in measurement.

1 The formulas used to calculate the circumference and area of a circle of radius r are examples of algebraic expressions.

[You can work in standard mode and tap , instead of pressing E to obtain decimal answers.]

• Working with the M application, tap Edit then Clear All Variables then tap OK and 2πr⇒c using the 9 keyboard. Press E. On the bottom toolbar, tap Standard to change the mode to decimal unless the calculator is already in decimal mode.

• Now repeat this process for the formula A = πr 2

• To substitute values into the defined expressions, use the U symbol located in the soft keyboard 9, then OPTN. To find the circumference and area of a circle of radius 10 cm, type ‘c|r = 10’ and ‘a|r = 10’.

a What is the circumference (correct to 1 decimal place)? cm (approx.)

b What is the area (correct to 1 decimal place)? cm 2

c Now find the radius of a circle with circumference of 20 cm, correct to 1 decimal place. Still using the M application, tap Action then Equation/Inequality then Solve. Type ‘solve(c = 20,r)’. Press E cm

2 Now define the simple interest earned, I, on an amount of money, P, that is invested at a certain interest rate, r%, for a certain time, T. Then use this formula to find the principal if the interest earned is $150, the interest rate is 4% and time the money is invested is 2 years.

[It is important to use the 9 italicised variables for b and h and not the b and h from the 0 keyboard if the multiplication signs are omitted.]

• Type ‘ prt 100 W i’. Press E

• Tap Action then Equation/inequality then solve. Type ‘i=150,p)|{r=4,t=2}’then press E. What is the amount of money invested?

3 Samantha went to JC Hi-Fi and bought 25 blank DVDs. She paid $28.75.

a If p is the cost of a single DVD, complete this equation: × p =

b Now use ‘solve (25p = 28.75, p)’ to find the cost of a single DVD. Cost per DVD = $

c Write an expression for the cost, c, of purchasing a number, n, of DVDs, each priced at $1.05.

d Now use your calculator to calculate the number of DVDs if the cost was $32.55.

Try this!

A tin can has volume V = πr 2h, where r is the radius of the base and h is the height of the can. Use your calculator to find the following (correct to 1 decimal place):

a volume of a tin can with radius 5 cm and height 10 cm

b radius of a tin with volume 100 cm 3 and height 5 cm

c height of a tin with volume 200 cm 3 and radius 3.5 cm.

Make sure that you do not delete your answers from the calculator.

9 Algebraic expressions

1 The formulas used to calculate the circumference and area of a circle of radius r are examples of algebraic expressions.

[When you move on to a new problem, it is a good idea to clear all variables used previously. This can be done by pressing b1 4.]

• Open a new document with a Calculator page. Press b11 to select Define from the Actions menu. Type ‘c = 2π × r’. Press ·

• Now repeat this process for the formula ‘A = πr 2’

• To substitute values into the defined expressions use the ñ (with) symbol. To find the circumference and area of a circle of radius 10 cm, type ‘c|r = 10’ and ‘a|r = 10’. In the Auto Mode, the calculator gives exact values. To find decimal approximations press /·

a What is the circumference (correct to 1 decimal place)? cm

b What is the area (correct to 1 decimal place)? cm 2

c Now find the radius of a circle with circumference of 20 cm. In the same calculator page press b31 to select Solve from the Algebra menu. Type ‘solve(c = 20, r)’. Press /· for an approximate answer. cm

[It is important to include × signs between pronumerals, each of which is shown as a . on the screen between the variables. Otherwise, the calculator treats bh as one variable.]

• Press b11 to select Define from the Actions menu. Type ‘ i = p r t 100 ’. The fraction template can be used by pressing /p. Press ·

• Type ‘solve (i=150, p)|r=4 and t=2’. The and command can be typed manually with a space on either side. What is the amount of money invested?

3 Samantha went to JC Hi-Fi and bought 25 blank DVDs. She paid $28.75.

a If p is the cost of a single DVD, complete this equation:

× p =

b Now use ‘solve (25p = 28.75, p)’ to find the cost of a single DVD. $

c Write an expression for the cost, c, of purchasing a number, n, of DVDs, each priced at $1.05

d Now use your calculator to calculate the number of DVDs if the cost was $32.55.

Try this!

A tin can has volume V = pr 2h where r is the radius of the base and h is the height of the can. Use your calculator to find the following (correct to 1 decimal place):

a volume of a tin can with radius 5 cm and height 10 cm

b radius of a tin with volume 100 cm 3 and height 5 cm

c height of a tin with volume 200 cm 3 and radius 3.5 cm.

Remember to save your calculator page!

Chance 1

1 A result obtained when an experiment is conducted is called an .

[Definition of terms]

2 The set of possible outcomes of a probability experiment is called a

[Definition of terms]

[Sample space]

3 A team plays 2 games. They can win (W) or lose (L) each game. No draws are allowed. Draw a tree diagram and use it to write the sample space for the possible outcomes of the two games.

A student rolled a die 10 times and tallied the results as shown. Use this data for 4–9

4 How many times did the student roll a 2?

[Trial]

5 What was the relative frequency of rolling a 2? Give the answer as a fraction.

[Relative frequency]

6 Which event(s) had a relative frequency of 1 5 ?

7 What was the experimental probability of rolling a 2?

[Experimental probability]

8 If the die is fair, what is the theoretical probability of rolling a 2?

[Theoretical probability]

9 If the number of trials (rolls) was increased to 1000, what would you expect to happen to the experimental probability?

For 10–15, one card is drawn at random from a standard deck of 52 cards.

[Theoretical probability]

10 What is the theoretical probability that the card is a spade? Give the answer as a simple fraction.

11 Find Pr(Ace) as a simple fraction.

[Theoretical probability]

12 Find Pr(Ace of spades) as a fraction.

[Theoretical probability]

13 Locate Pr(spade) on this probability scale.

[Theoretical probability]

14 Locate Pr(Ace) on the scale above.

[Theoretical probability]

15 Locate Pr(Ace of spades) on the scale above.

[Theoretical probability]

For 16–18, Bag 1 contains 5 counters numbered 1 to 5. Bag 2 contains four different-coloured marbles.

The following lattice diagram shows all possible outcomes if one item is selected at random from each bag.

17 Find Pr(white marble) as a simple fraction.

[Theoretical probability]

18 Find Pr(red marble and 5) as a decimal fraction.

[Theoretical probability]

19 Find Pr(yellow or red marble and a number less than 3) as a fraction.

[Theoretical probability]

20 Find the complement of the event in 19.

[Complementary events]

For 21–23, the table shows the number of CDs of each artist in stock at the Melbourne and Sydney Mighty Music Megastores.

Calculate each probability as a fraction in simplest form.

21 Pr(CD randomly selected by a customer is a The Mathematicians CD).

[Theoretical probability]

16 Find the total number of possible outcomes.

[Lattice diagram]

[Theoretical probability]

22 Pr(CD randomly selected by a customer at the Melbourne store is a Trigonometrics CD).