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

Recurring decimals, fractional powers, surds, bounds and counting

Grades 4–9About 85 minutesNo calculator90 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
Write $\frac{3}{8}$ as a decimal.
[1 mark]
Answer

A2
Write $5^{3} \times 5^{4}$ as a single power of 5.
[1 mark]
Answer
A3
Write 72 as a product of its prime factors. Give your answer in index form.
[2 marks]
Answer
A4
Round 7.0483 to 3 significant figures.
[1 mark]
Answer
A5
Write down the factors of 48 that are square numbers.
[1 mark]
Answer

Examples, then your turn

Grades 5–6

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

Example 1

Write $0. 2 ˙ 4 ˙$ as a fraction in its simplest form.

Hint Call the decimal

x

. Two digits repeat, so multiply by 100 and subtract

x

B1
Write $0. 6 ˙ 3 ˙$ as a fraction in its simplest form.
[2 marks]
Answer

Example 2

Work out the value of $2 7^{- \frac{2}{3}}$ .

Hint The negative sign means one over. The 3 on the bottom means a cube root, and the 2 on top means square it.

B2
Work out the value of $12 5^{- \frac{2}{3}}$ .
[2 marks]
Answer

Example 3

Simplify $112 - 28$ .

Hint Find the largest square factor of each number. Both surds become a multiple of

7

B3
Simplify $99 + 44$ .
[2 marks]
Answer

Example 4

A rectangle is 8.4 cm long and 5.2 cm wide. Both lengths are correct to 1 decimal place.

Work out the upper bound of the perimeter of the rectangle.

Hint Each length could be up to 0.05 cm more than given. A perimeter is a sum, so use the two upper bounds.

B4
A rectangle is 7.3 cm long and 4.6 cm wide. Both lengths are correct to 1 decimal place.
Work out the lower bound of the perimeter of the rectangle.
[2 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
Here are three fractions.
$\frac{7}{40} \frac{5}{12} \frac{9}{75}$
Which of them can be written as a terminating decimal? Give a reason for your answer.
[2 marks]
Answer

C2
Write $0.3 1 ˙ 8 ˙$ as a fraction in its simplest form.
[3 marks]
Answer
C3
Work out the value of $(\frac{64}{125})^{- \frac{2}{3}}$ .
[2 marks]
Answer
C4
Find the value of $x$ when $9^{x} = \frac{1}{27}$ .
[2 marks]
Answer $x = \dots\dots$
C5
Give each answer in its simplest exact form.
(a)
Expand and simplify $(3 + 5) (2 - 5)$ .
[2 marks]
Answer
(b)
Rationalise the denominator of $\frac{12}{6}$ .
[2 marks]
Answer
Total for C5: 4 marks
C6
The mass of a parcel is 64 kg, correct to the nearest kilogram.
Write down the error interval for the mass, $m$ kg.
[2 marks]
Answer $\dots\dots \leq m < \dots\dots$
C7
A number $y$ is truncated to 1 decimal place. The result is 7.2
Write down the error interval for $y$ .
[2 marks]
Answer $\dots\dots \leq y < \dots\dots$
C8
A car park ticket has a code made of two letters followed by three digits. The letters are from A to Z and the digits from 0 to 9. Letters and digits may be repeated.
How many different codes are possible?
[2 marks]
Answer
C9
Three-digit numbers are made from the digits 1, 2, 3, 4, 5 and 6. No digit is used more than once in a number.
How many of these numbers are even?
[2 marks]
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Prove algebraically that the recurring decimal $0.4 5 ˙$ can be written as $\frac{41}{90}$ .
[3 marks]

D2
Ali says, " $6 4^{- \frac{1}{2}} = - 8$ , because the minus sign makes the answer negative."
Ali is wrong. Explain why, and work out the value of $6 4^{- \frac{1}{2}}$ .
[2 marks]
Answer
D3
Solve $4^{x + 1} = 8^{x - 1}$ .
You must show your working.
[3 marks]
Answer $x = \dots\dots$
D4
Show that $(45 + 20)^{2} = 125$ .
[3 marks]
D5
$\frac{6}{2 - 3}$ can be written in the form $a + b 3$ , where $a$ and $b$ are integers.
Find the value of $a$ and the value of $b$ .
[4 marks]
Answer $a = \dots\dots$ $b = \dots\dots$
D6
The diagram shows a right-angled triangle. The two shorter sides are $(2 + 3)$ cm and $(2 - 3)$ cm.
Not drawn accurately
(a)
Show that the area of the triangle is $\frac{1}{2}$ cm².
[2 marks]
(b)
Work out the length of the longest side. Give your answer as a surd.
[3 marks]
Answer
(c)
Write the perimeter of the triangle in the form $a + b$ .
[1 mark]
Answer
Total for D6: 6 marks
D7
A bike lock opens with a code of four digits. Each digit is from 0 to 9.
(a)
How many different codes are possible?
[1 mark]
Answer
(b)
How many of the codes have four different digits?
[2 marks]
Answer
(c)
Kai's code has four different digits. It starts with 7 and ends with an even digit.
How many codes could Kai's code be?
[2 marks]
Answer
Total for D7: 5 marks
D8
Simplify fully $(27 x^{6} y^{- 3})^{\frac{2}{3}}$ .
[4 marks]
Answer
D9
A runner runs 70 m, correct to the nearest 10 m. She takes 13 seconds, correct to the nearest second.
(a)
Work out the upper bound of her average speed. You must show your working.
[3 marks]
Answer
(b)
Show that her average speed could have been less than 5 m/s.
[2 marks]
Total for D9: 5 marks
D10
Eight students stand for a club committee. One is chosen as captain, a different one as vice-captain and a third as secretary.
(a)
In how many different ways can the three jobs be filled?
[2 marks]
Answer
(b)
Ella must be the captain. In how many ways can the other two jobs be filled?
[1 mark]
Answer
Total for D10: 3 marks
D11
$2^{x} \times 4 = 8^{3}$
Find the value of $x$ .
[3 marks]
Answer $x = \dots\dots$

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Show that $0. 1 ˙ 4285 7 ˙ = \frac{1}{7}$ .
[4 marks]
Hint 1 · What to try
Six digits repeat. Multiply by a power of 10 that moves one whole block of them past the decimal point.
Hint 2 · The first line
$1000000 x - x = 142857$
Hint 3 · The full method
$x = \frac{142 857}{999 999}$ , and $7 \times 142857 = 999999$ , so $x = \frac{1}{7}$ .

E2
Write $\frac{5 + 1}{5 - 1}$ in the form $\frac{a + b}{c}$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Hint 1 · What to try
To clear a surd from a bracket on the bottom, multiply top and bottom by the same bracket with the sign changed.
Hint 2 · The first line
$\frac{( 5 + 1 ) ( 5 + 1 )}{( 5 - 1 ) ( 5 + 1 )} = \frac{6 + 2 5}{4}$
Hint 3 · The full method
Divide every term by 2: $\frac{3 + 5}{2}$ .
Answer
E3
How many three-digit numbers have digits that add up to 6?
[3 marks]
Hint 1 · What to try
Work through the first digit, from 1 to 6, one at a time.
Hint 2 · The first line
If the first digit is 1, the other two add to 5. There are 6 ways: 05, 14, 23, 32, 41, 50.
Hint 3 · The full method
The counts for first digits 1 to 6 are 6, 5, 4, 3, 2, 1. Their total is 21.
Answer
E4
Solve $3^{2 x} - 10 \times 3^{x} + 9 = 0$ .
[4 marks]
Hint 1 · What to try
$3^{2 x}$ is $(3^{x})^{2}$ . Give $3^{x}$ a letter of its own.
Hint 2 · The first line
With $y = 3^{x}$ : $y^{2} - 10 y + 9 = 0$ , so $(y - 1) (y - 9) = 0$ .
Hint 3 · The full method
$3^{x} = 1$ gives $x = 0$ , and $3^{x} = 9$ gives $x = 2$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can tell which fractions terminate and change a recurring decimal into a fraction.
I can work out negative and fractional powers, and solve equations with powers.
I can simplify surds, expand brackets with surds and rationalise a denominator.
I can write an error interval and choose the right bounds in a calculation.
I can count outcomes with the product rule and check by listing.

Answers and mark scheme

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