KS3 Maths · Year 8 · Chapter 2: Equations and formulae · Lesson 4

Equations with fractions

Year 8About 70 minutesNo calculator50 marks

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Date

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Quick questions to start.

A1
Needed today
Solve $\frac{x}{4} = 7$ .
[1 mark]
Answer
A2
Needed today
Solve $3 (x - 2) = 15$ .
[1 mark]
Answer
A3
Last lesson
Factorise $6 ab + 15 a$ as much as possible.
[1 mark]
Answer
A4
Year 7
Write $\frac{12}{30}$ in its simplest form.
[1 mark]
Answer

Example, then your turn

Year 8

When $x$ is on the bottom of a fraction, multiply both sides by the bottom first. A bracket on the bottom stays a bracket: multiply by all of it. Then solve as usual, and check by putting your answer back in. Your teacher works through each example with you. Then try the one beside it.

Example 1

Solve $\frac{18}{x} = 4$ .

Hint Multiply both sides by

x

B1
Solve (i) $\frac{35}{x} = 7$ (ii) $\frac{21}{x} = 6$ .
[2 marks]
Answer(i) $x = \dots\dots$ (ii) $x = \dots\dots$

Example 2

Solve $\frac{15}{4 x} = 3$ .

Hint Multiply both sides by

4 x

15 = 12 x

B2
Solve (i) $\frac{21}{2 x} = 6$ (ii) $\frac{8}{3 x} = 4$ .
[2 marks]
Answer(i) $x = \dots\dots$ (ii) $x = \dots\dots$

Example 3

Solve $\frac{36}{x + 3} = 4$ .

Hint Multiply both sides by

(x + 3)

, or say "

36

divided by what is

4

B3
Solve (i) $\frac{45}{x - 4} = 9$ (ii) $\frac{28}{2 x + 1} = 4$ .
[2 marks]
Answer(i) $x = \dots\dots$ (ii) $x = \dots\dots$

Example 4

Solve $\frac{x + 4}{x - 2} = 3$ .

Hint Multiply both sides by

(x - 2)

. Then you have

x

on both sides.

B4
Solve $\frac{2 x + 3}{x - 1} = 3$ .
[3 marks]
Answer $x = \dots\dots$

Example 5

$N$ is a number. Three times $N$ , divided by $2$ less than $N$ , equals $4$ . Write an equation and solve it to find $N$ .

Hint "

2

less than

N

" is

N - 2

, and it is the whole bottom of the fraction.

B5
$x$ is a number. The ratio $x : (x + 6)$ is equal to $3 : 5$ . Write an equation with fractions and solve it to find $x$ .
[3 marks]
Answer $x = \dots\dots$

Practice

Year 8

C1
Solve $\frac{12}{x + 7} = 3$ .
[2 marks]
Answer $x = \dots\dots$
C2
Solve $\frac{30}{10 - x} = 6$ .
[2 marks]
Answer $x = \dots\dots$

C3
Kai says that $x = 2$ is the solution of $\frac{14}{3 x - 1} = 2$ . Is Kai right? Show how you decide.
[2 marks]
C4
Speed $=$ distance $\div$ time. A train travels $210$ km at an average speed of $56$ km/h. Use the formula to write an equation for the time $t$ hours, and solve it.
[2 marks]
Answer $t = \dots\dots$ hours
C5
Solve $\frac{5}{2 x} = \frac{1}{4}$ .
[2 marks]
Answer $x = \dots\dots$
C6
$60$ divided by a number is $4$ more than $8$ . Write an equation and solve it to find the number.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Ali solves $\frac{20}{x + 3} = 4$ . He writes $20 = 4 x + 3$ , and gets $x = 4.25$ .
Ali is wrong. Explain his mistake, and solve the equation correctly.
[3 marks]
Answer $x = \dots\dots$

D2
A rectangle is $6$ cm wide and $(2 x - 4)$ cm long. Its area is $72$ cm².
Not drawn accurately
(a)
Explain why $\frac{72}{2 x - 4} = 6$ .
[1 mark]
(b)
Solve the equation to find $x$ .
[2 marks]
Answer $x = \dots\dots$
(c)
Work out the perimeter of the rectangle.
[2 marks]
Answer
Total for D2: 5 marks
D3
Solve $\frac{8}{x + 1} = \frac{6}{x - 1}$ .
[3 marks]
Answer $x = \dots\dots$
D4
A car travels $150$ km at an average speed of $(x + 10)$ km/h. The journey takes $2.5$ hours. Write an equation and solve it to find $x$ .
[3 marks]
Answer $x = \dots\dots$
D5
There are $x$ numbers in a list. Their total is $4 x + 18$ . The mean of the numbers is $6$ .
(a)
Write an equation in $x$ .
[1 mark]
Answer
(b)
Solve your equation to find how many numbers there are.
[2 marks]
Answer
(c)
Check your answer by working out the total and the mean.
[1 mark]
Total for D5: 4 marks

Extension

Stretch

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

E1
The same whole number is added to the top and to the bottom of the fraction $\frac{3}{8}$ . The new fraction is equal to $\frac{2}{3}$ . What number was added?
[4 marks]
Hint 1 · What to try
Call the number $n$ . Write the new fraction with $n$ in it.
Hint 2 · The first line
$\frac{3 + n}{8 + n} = \frac{2}{3}$ . Multiply both sides by $3 (8 + n)$ .
Hint 3 · The full method
$3 (3 + n) = 2 (8 + n)$ , so $9 + 3 n = 16 + 2 n$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve an equation with $x$ on the bottom of a fraction.
I can solve an equation with a bracket on the bottom, or $x$ on the top and the bottom.
I can solve an equation with a fraction on each side by cross-multiplying.
I can write and solve a fraction equation from words, a ratio or a formula.

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