GCSE Maths | Ratios
Simple Ratios
Worked Example
Share £40 in the ratio 3:5
$3 : 5 \quad (8)$
$? : ? \quad (40)$
$sf = 40 \div 8 = 5$
$3 \times 5 = 15, \quad 5 \times 5 = 25$
Question 1
a) Share £20 in the ratio 2:3
b) Share 15cm in the ratio 1:2
c) Divide £24 in the ratio 1:3
d) Share 35 sweets in the ratio 4:3
e) Divide 55g in the ratio 3:2
f) Divide 54kg in the ratio 1:5
g) Share £104 in the ratio 3:5
h) Divide 161 miles in the ratio 6:1
i) Divide 315ml in the ratio 2:7
j) Share $650 in the ratio 4:9
k) Share £800 in the ratio 11:14
l) Share 1200kg in the ratio 3:37
Show Answers
a) £8, £12
b) 5cm, 10cm
c) £6, £18
d) 20, 15 sweets
e) 33g, 22g
f) 9kg, 45kg
g) £39, £65
h) 138, 23 miles
i) 70ml, 245ml
j) $200, $450
k) £352, £448
l) 90kg, 1110kg
Applying Ratios
Worked Example
The ratio of cats to dogs in a shelter is 3:4. There are 12 cats. How many dogs are there?
$3 : 4$
$12 : ?$
$sf = 12 \div 3 = 4$
Dogs $= 4 \times 4 = 16$
Question 2
a) James has some apples and oranges. The ratio of apples to oranges is 2:5. He has 15 oranges. How many apples does James have?
b) The ratio of lemon sweets to strawberry sweets in a tub is 5:3. There are 120 lemon sweets in the tub. How many strawberry sweets are in the tub?
c) Rachel has some first class and some second class stamps. The ratio of first class to second class stamps is 3:4. Rachel has 18 first class stamps. (i) How many second class stamps does Rachel have? (ii) How many stamps does Rachel have in total?
d) Abby, Neil and Dylan share a sum of money in the ratio 2:4:5. Neil receives £60. Work out how much money Dylan receives.
e) Isaac and Victoria share money in the ratio 2:5. Victoria receives £6.60. Work out the difference between how much money Isaac and Victoria receive.
f) Rachel has some apples and bananas. The ratio of apples to bananas is 2:3. She has 14 apples. Work out how many bananas Rachel has.
g) Chris and Molly win money in a competition. They share the money in the ratio 2:3. Molly receives £240. How much money does Chris receive?
h) The ratio of boys to girls in a school is 4:5. There are 220 boys in the school. How many students attend the school?
i) Three angles are in the ratio 2:3:5. The smallest angle is 50°. Work out the sizes of the other two angles.
Show Answers
a) 6 apples
b) 72 strawberry sweets
c) (i) 24 second class stamps (ii) 42 stamps in total
d) £75
e) £3.96
f) 21 bananas
g) £160
h) 495 students
i) 75° and 125°
Comparing Ratios
If you know the difference between two shares, that difference corresponds to a whole number of parts. Divide the difference by that number of parts to find one share.
Worked Example
Sam and Tia share sweets in the ratio 2:5. Tia gets 18 more sweets than Sam. How many sweets does each person get?
Difference in parts $= 5-2 = 3$
One share $= 18 \div 3 = 6$
Sam $= 2 \times 6 = 12$, Tia $= 5 \times 6 = 30$
Question 3
a) Charlene and Danielle share some money in ratio 2:3. Danielle gets £25 more than Charlene. How much does each girl receive?
b) The costs of building a single-storey house extension are in the ratio labour : materials : professional fees $= 7:5:1$. The cost of materials is £6,400 more than the amount paid in professional fees. Work out the total cost of building this house extension.
c) Three quantities $A$, $B$ and $C$ are in the ratio $A:B:C$ is $13:7:2$. Given that $A - B = 48$, find the value of $A + B + C$.
d) Last season, the number of goals scored by Arjun, by Simon and by Kath for their football team were in the ratios 2:5:8. Simon scored 12 more goals than Arjun. Work out the number of goals scored by Kath.
Show Answers
a) Charlene: £50, Danielle: £75
b) £20,800
c) $A+B+C = 176$
d) $32$ goals
Combining Ratios
Worked Example
$a:b = 2:3$ and $b:c = 6:5$. Find $a:b:c$.
$a:b = 4:6$ and $b:c = 6:5$
$a:b:c = 4:6:5$
Question 4
Combine the following ratios, or find $a:b:c$.
a) $a:b$ is $3:2$ and $b:c$ is $2:5$
b) $a:b$ is $3:1$ and $b:c$ is $2:3$
c) $a:b$ is $7:4$ and $b:c$ is $2:1$
d) $a:b$ is $2:3$ and $b:c$ is $2:5$
e) $a:b$ is $9:2$, $b:c$ is $6:7$, and $a+b+c = 200$. Find $a$, $b$ and $c$.
Show Answers
a) $a:b:c = 3:2:5$
b) $a:b:c = 6:2:3$
c) $a:b:c = 7:4:2$
d) $a:b:c = 4:6:15$
e) $a = 135$, $b = 30$, $c = 35$
Ratios and Equations
Worked Example (ratio to equation)
$a:b = 4:7$
$7a = 4b$
Worked Example (equation to ratio)
$3a = 8b$
$a:b = 8:3$
Question 5
Turn each ratio into an equation.
a) $a:b = 6:5$
b) $a:b = 3:4$
c) $a:b = 7:2$
d) $a:b = 5:9$
e) $a:b = 8:3$
Turn each equation into a ratio $a:b$.
f) $6a = 5b$
g) $4a = 9b$
h) $3a = 7b$
i) $2a = 5b$
j) $5a = 8b$
Show Answers
a) $5a = 6b$
b) $4a = 3b$
c) $2a = 7b$
d) $9a = 5b$
e) $3a = 8b$
f) $a:b = 5:6$
g) $a:b = 9:4$
h) $a:b = 7:3$
i) $a:b = 5:2$
j) $a:b = 8:5$
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