year 8 maths

34 Chapter 1 Algebraic techniques 2 and indices
PROBLEM-SOLVING AND REASONING
11, 15a,b 12, 14–16
12–15, 17, 18
11 Write an expression for each of the following and then expand it.
a A number t has 4 added to it and the result is multiplied by 3 .
b A number u has 3 subtracted from it and the result is doubled.
c A number v is doubled, and then 5 is added. The result is tripled.
d A number w is tripled, and then 2 is subtracted. The result is doubled.
12 Match each operation on the left with an equivalent one on the right. (Hint: first convert the
descriptions into algebraic expressions.)
a The number x is doubled and 6 is added. A x is doubled and reduced by 10 .
b The number x is reduced by 5 and the
B The number x is tripled.
result is doubled.
c The number x is added to double the value C x is decreased by 6 .
of x .
d The number x is halved, then 3 is added
and the result is doubled.
D x is increased by 3 and the result is
doubled.
e 2 is subtracted from one-third of x and the
result is tripled.
E x is increased by 6 .
13 The number of boys in a classroom is b and the number of girls is g . Each boy has five pencils and
each girl has three pencils.
a Write an expression for the total number of pencils in the class.
b If the pencils cost $2 each, write and expand an expression for the total cost of all the pencils in
the room.
c Each boy and girl also has one pencil case, costing $4 each. Write a simplified and expanded
expression for the total cost of all pencils and cases in the room.
d If there are 10 boys and eight girls in the room, what is the total cost for all the pencils and
cases in the room?
14 a When expanded, 4(2a + 6b) gives 8a + 24b . Find two other expressions that expand to give
8a + 24b .
b Give an expression that expands to 4x + 8y .
c Give an expression that expands to 12a − 8b .
d Give an expression that expands to 18ab + 12ac .
15 The distributive law is often used in multiplication of whole numbers. For example:
17 × 102 = 17 × (100 + 2) = 17(100) + 17(2) = 1734
a Use the distributive law to find the value of 9 × 204 . Start with 9 × 204 = 9 × (200 + 4) .
b Use the distributive law to find the value of 204 × 9 . Start with 204 × 9 = 204 × (10 − 1) .
c Given that a × 11 = a × (10 + 1) = 10a + a , find the value of these multiplications.
i 14 × 11 ii 32 × 11 iii 57 × 11 iv 79 × 11
d It is known that (x + 1)(x − 1) expands to x 2 − 1 . For example, if x = 7 this tells you that
8 × 6 = 49 − 1 = 48 . Use this fact to find the value of:
i 7 × 5 ii 21 × 19 iii 13 × 11 iv 201 × 199
e Using a calculator, or otherwise, evaluate 15 2 , 25 2 and 35 2 . Describe how these relate to the fact
that (10n + 5)(10n + 5) is equivalent to 100n(n + 1) + 25 .
Cambridge Maths NSW
Stage 4 Year 8 Second edition
ISBN 978-1-108-46627-1 © Palmer et al. 2018
Cambridge University Press
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