GCSE Maths · Higher · Chapter 16: Counting, accuracy, powers and surds · Lesson 1

Rational numbers and recurring decimals

Grades 5–8About 45 minutesNo calculator42 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out $\frac{2}{5} + \frac{1}{3}$ .
[1 mark]
Answer
A2
From chapter 7
In triangle $A B C$ , $A B = 8$ cm, $B C = 5$ cm and angle $B = 4 8^{\circ}$ . In triangle $P QR$ , $P Q = 8$ cm, $QR = 5$ cm and angle $Q = 4 8^{\circ}$ . Which condition shows they are congruent?
[1 mark]
Answer
A3
From chapter 14
Write $7.1 \times 1 0^{- 2}$ as an ordinary number.
[1 mark]
Answer
A4
Last lesson
Between which two whole numbers is the solution of $x^{3} = 100$ ?
[1 mark]
Answer

Grades 6–7

Your teacher works through each example with you. Then try the one beside it.

Example 1

Write $0. 3 ˙ 6 ˙$ as a fraction in its simplest form.

Hint Two digits repeat, so multiply by 100 and subtract.

Example 2

Write $0.1 6 ˙$ as a fraction in its simplest form.

Hint Only the 6 repeats. Line up the repeats with

100 x

and

10 x

Grades 6–7

C1
Which of $\frac{9}{32}$ , $\frac{7}{30}$ and $\frac{21}{35}$ give a terminating decimal? Give a reason.
[2 marks]
Answer
C2
Write $0.124$ as a fraction in its simplest form.
[2 marks]
Answer

C3
Write $0. 4 ˙ 0 5 ˙$ as a fraction in its simplest form.
[3 marks]
Answer
C4
Write down the reciprocal of $3 \frac{1}{3}$ .
[1 mark]
Answer
C5
Write $\frac{7}{27}$ as a recurring decimal. Use dot notation.
[2 marks]
Answer
C6
Write $3. 2 ˙ 4 ˙$ as a mixed number in its simplest form.
[3 marks]
Answer
C7
Which of these numbers are irrational? $25$ , $18$ , $0. 3 ˙$ , $π$ , $\frac{22}{7}$
[2 marks]
Answer
C8
Write these in order of size, smallest first. $0. 6 ˙ 7 ˙$ , $0.67$ , $0. 6 ˙$ , $0.66$
[2 marks]
Answer

Grades 7–8

D1
Prove algebraically that the recurring decimal $0.2 3 ˙ 6 ˙$ can be written as $\frac{13}{55}$ .
[3 marks]

D2
Mia says, " $\frac{7}{56}$ gives a recurring decimal, because 56 has a prime factor of 7."
Explain why Mia is wrong.
[2 marks]
D3
$x = 0. 7 ˙ 2 ˙$ and $y = 0. 2 ˙ 7 ˙$
(a)
Write $x$ as a fraction in its simplest form.
[2 marks]
Answer
(b)
Write $y$ as a fraction, and hence show that $x + y = 1$ .
[3 marks]
Total for D3: 5 marks
D4
Work out the reciprocal of $0.1 2 ˙$ . Give your answer as a mixed number.
[3 marks]
Answer

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can decide whether a fraction gives a terminating or a recurring decimal.
I can prove algebraically what fraction a recurring decimal is.
I can find a reciprocal and tell a rational number from an irrational one.

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