1. jfrost@tiffin.kingston.sch.uk @DrFrostMaths
GCSE :: Ratio
@DrFrostMaths
Objectives: Understanding what is meant by ratio. Solve problems where a value and a ratio is given. Combine ratios. Solve problems involving changing ratios.
Last modified: 29th December 2018

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3. Overview
Ratio allows us to express the ‘relative size’ of different quantities.
Lesson 1: Simplifying ratios and solving problems where one amount or total is given.
Lesson 2: Combining or splitting ratios.
“The ratio of red to blue counters is 3:5 and ratio of blue to green counters 6:7. What is the ratio of red to green counters?”
“The ratio of muffins to cupcakes in a cake sale is 3:1. There’s 20 cakes in total. How many muffins are there?”
Lesson 3: Further Ratio Problem Solving (Advanced)
“Javier and Carmen share some money in the ratio 5 : 6. When each are given £4 more, the ratio of the money they have is now 6 : 7. How much money did each initially have?”

4. Red 18 Blue 12 Stand up if… You have two colours to pick from only.
Stand up if you prefer (a) Red (b) Blue
Suppose the results were the following:
Suppose also we were unconcerned by exact numbers who voted for each but wanted to express the relative size of how many voted for each. What ways could we do this?
Red 18
Blue 12
Percentages: ?
Red 60%, Blue 40%
Percentages are good for indicating proportion of some total.
Ratio of red to blue:
3 : 2
Ratio: ?
(We’ll explore this on the next slide…)

5. “The ratio of people who chose red to blue is 3:2.”
What is ratio?
“The ratio of people who chose red to blue is 3:2.”
What specifically does this mean?
For each three people who chose red, two chose blue.
It could be that we just had 3 red people and 2 blue. The above statement would be true!
If we had 6 red and 4 blue, then it’s still the case that for each 3 red, there are 2 blue.
So 6:4=3:2
And similarly we could have 9 red and 6 blue.
This means we can times (or divide) each part in the ratio by the same number, and end up with an equivalent ratio.
3:2 = 6:4 = 9:6

6. Simplifying ratios
This suggests we can ‘simplify’ ratios in the same way as fractions.
Just like with fractions, we prefer the numbers to be as small as possible. ÷5 a ?
Simplify the following ratios:
10:5
12:18
14:21:28
10:5=2:1
12:18=2:3
14:21:28=2:3:4 ÷6 b ? ÷7 c ?
Quickfire Simplifying:
6:3=𝟐:𝟏 12:16=𝟑:𝟒 10:30=𝟏:𝟑 14:2:8=𝟕:𝟏:𝟒 16:24:40=𝟐:𝟑:𝟓 18:6:30=𝟑:𝟏:𝟓 ?
Alice has £30 and Bob has £35. What is the ratio of the money Alice has to what Bob has?
𝟑𝟎:𝟑𝟓 → 𝟔:𝟕 a g ? b c ? d ? ? ? e f ?

7. Splitting an amount into a given ratio
The ratio of muffins to cupcakes in a cake sale is 3:1. There’s 20 cakes in total. How many muffins are there?
Method 1: Identify what ‘1 part’ is worth
Method 2: The ‘scaling’ way ?
In the ratio 3:1, we have “3 parts muffin” to “1 part cupcake”.
For each 3 muffins we have 1 cupcake. That is 4 cakes.
But we want 20 cakes. So we have 5 lots of this, giving us 3×5=15 muffins and 1×5=5 cupcakes.
The given value in the question is the 20 cakes. How many ‘parts’ does this represent? ?
4 parts = 20
1 part = 5
3 parts = 15
There are 15 cupcakes. ? ?

8. Test Your Understanding
Method 1: Identify what ‘1 part’ is worth
Method 2: The ‘scaling’ way
(The one value given is the €56, which is the total amount Jack and Kate share. So how many parts does this represent?) ?
For each €5 Jack gets, Kate gets €9 and Lila €6. Jack and Kate totals €14.
€56 is 56÷14=4 times this.
Lila gets 6×4= € 24 ?
14 parts = €56
1 part = €4
6 parts = €24
Lila gets €24

9. When one of the quantities is given
In a death match of monkeys vs squirrels, the monkeys outnumber the squirrels by a ratio of 5:2. There’s 35 monkeys. How many squirrels are there?
Method 1: Identify what ‘1 part’ is worth
Method 2: The ‘scaling’ way ?
The one value given is the 35 monkeys. How many parts in the ratio does this represent?
For each 5 monkeys there are 2 squirrels.
There are 35 monkeys, which is 7 times as much. Thus there are also 7 times as many squirrels.
2×7=14 ?
5 parts = 35
1 part = 7
2 parts = 14
Therefore 14 squirrels.

10. When the difference is given
Alice and Bob share some money in the ratio 7:4.
Alice received £12 more than Bob.
What did they receive in total?
Method 1: Identify what ‘1 part’ is worth ?
This time £12 represents the DIFFERENCE in the parts.
3 parts = £12
1 part = £4
11 parts = £44
We want the total amount, so need to find the value of the total parts.
Method 2: The ‘scaling’ way
If Alice had £7 Bob would have £4. The difference is £3 but we want a £12 difference, so scale everything by 4.
Alice gets 7×4=28 and Bob 4×4=16. Total 28+16=44. ?

11. Test Your Understanding1
The ratio of cats to dogs at Battersea Dogs & Cats home is 2:3. There are 45 animals in total. How many cats are there? 2
The ratio of Tiffin Boys students to Tiffin Girls students is 8:9. There’s 2700 students at Tiffin Girls. How many at Tiffin Boys? ?
2400 ? 18 4
The ratio of the angles in a triangle is 4:3:2. Determine the largest angle. 3
Tom, Dick and Harry share some money in the ratio 4:7:2.
Dick receives £35 more than Harry. How much did Tom and Dick receive in total?
9 𝑝𝑎𝑟𝑡𝑠=180° 1 𝑝𝑎𝑟𝑡=20° 4 𝑝𝑎𝑟𝑡𝑠=80°
Largest angle is 80°
Sometimes you need to use your knowledge of what the values should add up to. The total angle in a triangle is 180°, and total probability is 1. ? ?
£77

12. Ratios to/from equations or worded descriptions
The ratio of red balls to yellow balls is 1:3. If the number of red balls is 𝑥 and the number of yellow balls 𝑦, write an equation for 𝑥 in terms of 𝑦.
The ratio of red balls to yellow balls is 3:5. If the number of red balls is 𝑥 and the number of yellow balls 𝑦, write an equation for 𝑥 in terms of 𝑦. ? ?
The number of red balls (𝒙) is 𝟑 𝟓 of the number of yellow balls (𝒚) so:
𝒙= 𝟑 𝟓 𝒚
The number of red balls (𝒙) is a third of the number of yellow balls (𝒚) so:
𝒙= 𝟏 𝟑 𝒚
Fro Tip: It might be helpful to say there’s actually 3 red balls and 5 yellow balls. Then it’s easier to see that 3 is of 5.
The number of yellow balls is twice the number of red balls and the number of green balls is 4 times the number of yellow balls. Determine the ratio of red to yellow to green balls. ?
If there was 1 red ball, there would be 2 yellow balls, and therefore 8 green balls. Ratio is therefore 𝟏:𝟐:𝟖
(We’d obtain the same ratio for any other starting number of red balls)

13. Exercise 1 ? ? ? ? ? ? ? ? ? ? ? ? (on provided worksheet)
Simplify the following ratios:
2:6 = 1: :15 = 2:3
24:18 = 4: : 49 = 8:7
Simplify the following:
8:10:12=𝟒:𝟓:𝟔 :14:35=𝟗:𝟐:𝟓
120 sweets are shared out into three piles, in the ratio 3:4:1. How many more sweets does the biggest pile have than the smallest?
The ratio of boys to girls at a birthday party is 4:5. If there are 35 girls, how many boys are there? 28
The ratio of boys to girls at another party is 5:7. There are 8 more girls than boys. How many boys are there? 20
The ratio of the probability of heads to probability of tails on an unfair coin is 1:4. What is the probability of heads?
𝟏 𝟓
The ratio of the angles in a quadrilateral are 7:4:5:2. What is the smallest angle?
𝟒𝟎° 1 6 a ? b ? c ? d ? ? 2 a 7 b 3
What is the ratio of shaded to non-shaded (small) triangles?
2 : 6 = 1 : 3 ? 8 ?
If £30 is split between two people in the ratio 3:2, what does each get? £18, £12
To make orange squash I mix concentrate and water in the ratio 2:7. If I use 150ml of concentrate, how much water do I use?
525ml ? 4 9 ? 5 ? 10 ? ?

14. Exercise 1 ? ? ? ? ? ? ? (on provided worksheet)
[JMC 2012 Q17] There are six more girls than boys in Miss Spelling’s class of 24 pupils. What is the ratio of girls to boys in this class?
5:3
[JMC 2000 Q9] Three –quarters of the junior members of a tennis club are boys and the rest are girls. What is the ratio of boys to girls among these members? A 3:4 B 4:3 C 3:7
D 4:7 E 3:1 Solution: E
[JMO 2012 A8] An athletics club has junior (i.e. boy or girl) members and adult members. The ratio of girls to boys to adults is 3 : 4 : 9 and there are 16 more adult members than junior members. In total, how many members does the club have? Solution: 128 14
If 18:𝑥=𝑥:8, find 𝑥.
𝒙=𝟏𝟐
The ratio of red to blue to green marbles in a bag is 3 : 10 : 4. If there are 39 red and blue marbles altogether, how many green marbles are there? Solution: 12
[IMC 2009 Q9] Joseph’s flock has 55% more sheep than goats. What is the ratio of goats to sheep in the flock? Solution: 20:31
[JMC 2012 Q19] In rectangle 𝑃𝑄𝑅𝑆, the ratio of ∠𝑃𝑆𝑄 to ∠𝑃𝑄𝑆 is 1:5. What is the size of ∠𝑄𝑆𝑅? A 15° B 18° C 45° D 72° E 75° Two angles add to 𝟗𝟎°. So 6 parts = 90  1 part = 𝟏𝟓°. Thus ∠𝑸𝑺𝑹=𝟗𝟎−𝟏𝟓°=𝟕𝟓° 11 ? 15 ? 12 ? 16 ? ? 13 17 ? ?

15. 𝒂 :𝒃 :𝒄 2 :3 4 :5 8 :12 :15 Combining and Subdividing Ratios
One new type of question to the GCSE syllabus is combining two ratios into one:
The ratio of 𝑎 to 𝑏 is 2:3. The ratio of 𝑏 to 𝑐 is 4:5. What is the ratio 𝑎:𝑏:𝑐?
𝒂 :𝒃 :𝒄
Tip: Give each variable a column. To be able to combine the ratios into one, we need the 𝑏’s to be the same. But we know we can scale ratios!
2 :3
4 :5 ×4 ×4 ×3 ×3
8 :12 :15

16. 𝒓 :𝒃 :𝒈 3 :5 6 :7 18 :30 :35 Further Example ?
The ratio of red to blue counters is 3:5 and ratio of blue to green counters 6:7. What is the ratio of red to green counters? ?
𝒓 :𝒃 :𝒈
3 :5
6 :7 ×6 ×5
18 :30 :35
So ratio of red to green is 18:35

17. Test Your Understanding? 24 15 20

18. Subdividing a Ratio ? ? ? All students study either French or Spanish.
The ratio of boys to girls is 3:4. Of the girls, the ratio studying French to those studying Spanish is 2:1.
What fraction of the students are girls studying Spanish?
The easiest method is to turn the ratios in fractions.
What fraction of students are girls? 4 parts out of 7  𝟒 𝟕
What fraction of girls study Spanish? 1 part out of 3  𝟏 𝟑
Therefore what fraction of the students are girls studying Spanish? of the of the students are Spanish-speaking girls. 𝟏 𝟑 𝒐𝒇 𝟒 𝟕 = 𝟏 𝟑 × 𝟒 𝟕 = 𝟒 𝟐𝟏 ? ? ?
Recall that we can replace the word ‘of’ with × when dealing with fractions of amounts.

19. Test Your Understanding?
Emma and Dave shared 𝟕 𝟏𝟎 of the money. Of this 𝟕 𝟏𝟎 , Dave gets 𝟐 𝟓 of it.
𝟐 𝟓 𝒐𝒇 𝟕 𝟏𝟎 = 𝟐 𝟓 × 𝟕 𝟏𝟎 = 𝟏𝟒 𝟓𝟎 = 𝟕 𝟐𝟓

20. Exercise 2 ? ? ? ? ? ? ? ? ? ? (on provided worksheet) 4 1 a b c d e 2
[Edexcel GCSE(9-1) Mock Set 3 Autumn F Q24b, 2H Q6b] Only blue vans and white vans are made in a factory. The ratio of the number of blue vans to the number of white vans is 4 : 3 For blue vans, the number of small vans : the number of large vans = 3 : 5 Work out the fraction of the number of vans made in the factory that are blue and large. Solution: 𝟐𝟎 𝟓𝟔
[JMC 2006 Q23] At a holiday camp, the ratio of boys to girls is 3:4 and the ratio of girls to adults is 5:7. What is the ratio of children to adults at the camp? A 4:5 B 5:4 C 12:7
D 15: E 21:20
35 : 28 = 5 : 4
The ratio of cats to dogs at a rescue centre is 2 : 3. Of the cats, 25% are black and of the dogs, 50% are black. What percentage of the total animals are black?
𝟏 𝟒 × 𝟐 𝟓 + 𝟏 𝟐 × 𝟑 𝟓 = 𝟐 𝟓 =𝟒𝟎%
(Note that it may have been easier to start with an actual number of animals, say 100, and go from there)
For each of the following, determine the ratio 𝑎:𝑐.
𝑎:𝑏=1:2, 𝑏:𝑐=3:4 → 𝟑:𝟖 𝑎:𝑏=4:3, 𝑏:𝑐=5:2 →𝟏𝟎:𝟑 𝑎:𝑏=1:2, 𝑏:𝑐=1:2 →𝟏:𝟒 𝑎:𝑏=4:3, 𝑏:𝑐=4:3 →𝟏𝟔:𝟗 𝑎:𝑏=11:2, 𝑏:𝑐=7:11 →𝟕:𝟐
The ratio of boys to girls at a party is 1 : 2. Of the boys, a quarter brought a present with them. What fraction of attendees are boys who brought a present?
𝟏 𝟒 × 𝟏 𝟑 = 𝟏 𝟏𝟐
[JMC 2010 Q10] At the Marldon Apple-Pie-Fayre bake-off, prize money is awarded for 1st, 2nd and 3rd places in the ratio 3:2:1. Last year Mrs Keat and Mr Jewell shared third prize equally. What fraction of the total prize money did Mrs Keat receive?
𝟏 𝟐 × 𝟏 𝟔 = 𝟏 𝟏𝟐 1 ? a b ? c ? d ? e ? ? 2 5 ? ? 3 6 ? ?

21. Exercise 2 ? ? ? ? ? (on provided worksheet) 7 9 8 10
[Edexcel GCSE(9-1) Mock Set 2 Spring F Q23, 2H Q2] On a school trip the ratio of the number of teachers to the number of students is 1: 15 The ratio of the number of male students to the number of female students is 7: 5 Work out what percentage of all the people on the trip are female students. Give your answer correct to the nearest whole number.
Solution: 39%
[JMC 2013 Q19] A swimming club has three categories of members: junior, senior, veteran. The ratio of junior to senior members is 3:2 and the ratio of senior members to veterans is 5:2. Which of the following could be the total number of members in the swimming club? A 30 B 35 C 48
D 58 E 60 Solution: D 9
[Edexcel GCSE(9-1) Mock Set 3 Autumn H Q11] Anna and Bill share some money in the ratio 2 : 5 Anna gets £𝐴 Bill gets £𝐵 Carl and Donna share twice as much money as Anna and Bill share. They share the money in the ratio 3 : 1 Carl gets £𝐶 and Donna gets £𝐷. Find 𝐴: 𝐵: 𝐶:𝐷, giving your answer in its simplest form.
Solution: 𝟒:𝟏𝟎:𝟐𝟏:𝟕
Shapes are black and white, and are circles or squares.
The ratio of circles to squares is 2:3. Of the circles, the ratio of black to white is 4:5. Of the squares, the ratio of black to white is 5:2 .
(a) What fraction of the shapes are black? 𝟐 𝟓 × 𝟒 𝟗 + 𝟑 𝟓 × 𝟓 𝟕 = 𝟏𝟗𝟏 𝟑𝟏𝟓
(b) Determine the ratio of black circles to white squares. 𝟖 𝟒𝟓 : 𝟔 𝟑𝟓 → 𝟐𝟖:𝟐𝟕 ? 8 ? 10 ? ? ?

22. Updated Ratios after Changing Amounts?
5 parts = 20 counters
1 part = 4 counters
So initially 4 red and 20 yellow.
We’re adding red counters but the number of yellow counters remains the same.
If new ratio 𝟏:𝟐, 2 parts = 20 counters, so 1 part = 10 counters.
So red counters added =𝟏𝟎−𝟒=𝟔

23. A few new ratio skills…
Before we move on to harder problems, here are two new key skills that will help us tackle them:
The ratio of the width to height of a TV is 16 : 9. How could we represent all possible widths and heights of the TV, that ensures this ratio is maintained? ?
The width and height could be 16cm and 9cm. But it could also be any multiple of this, e.g. 32cm and 18cm. So generally, we could multiply each of 16 and 9 by anything provided that it’s the same number: let’s call this 𝑥.
The width and height can be represented as 𝟏𝟔𝒙 and 𝟗𝒙.
If 𝑎:𝑏=𝑐:𝑑, form an equation relating 𝑎, 𝑏, 𝑐 and 𝑑. ?
Let’s look at an example:
𝟑:𝟒=𝟑𝟎:𝟒𝟎
Notice that 𝟑𝟎 𝟑 = 𝟒𝟎 𝟒 . Therefore more generally, if 𝒂:𝒃=𝒄:𝒅 we could write 𝒄 𝒂 = 𝒅 𝒃

24. Updated Ratios after Changing Amounts (harder)
Jamie and Alastair share some sweets in the ratio 7 : 5. Jamie gives 2 sweets to Alastair. The ratio of sweets is now 13 : 11. How many sweets did each initially have?
Trial and Error-ey but easiest method:
We could just scale each of the ratios until we see a pair of scalings that work… ?
Possible numbers of sweets before: (multiples of 7 : 5)
Jamie Alastair
7 5
14 10
21 15
28 20
35 25 …
Possible numbers of sweets after: (using 13 : 11)
Jamie Alastair
13 11
26 22
At this stage we can see that we could have started with 28 and 20 sweets, and after the exchange, each would have 26 and 22, as above.

25. Updated Ratios after Changing Amounts (harder)
Jamie and Alastair share some sweets in the ratio 7 : 5. Jamie gives 2 sweets to Alastair. The ratio of sweets is now 13 : 11. How many sweets did each initially have?
But what if we wanted to use an algebraic method?
Let’s try and use the two new skills we have just learnt. ?
Let Jamie have 𝟕𝒙 sweets and Alastair have 𝟓𝒙 sweets.
After Jamie gives 2 sweets to Alastair, Jamie has 𝟕𝒙−𝟐 and Alastair has 𝟓𝒙+𝟐.
Therefore:
𝟏𝟑 :𝟏𝟏 = 𝟕𝒙−𝟐 :𝟓𝒙+𝟐
𝟕𝒙−𝟐 𝟏𝟑 = 𝟓𝒙+𝟐 𝟏𝟏
𝟏𝟏 𝟕𝒙−𝟐 =𝟏𝟑 𝟓𝒙+𝟐 𝟕𝟕𝒙−𝟐𝟐=𝟔𝟓𝒙+𝟐𝟔 𝟏𝟐𝒙=𝟒𝟖 𝒙=𝟒
Therefore Jamie had 𝟕×𝟒=𝟐𝟖 sweets and Alastair had 𝟓×𝟒=𝟐𝟎
If 𝑎:𝑏=𝑐:𝑑 then 𝑐 𝑎 = 𝑑 𝑏
Cross-multiply.

26. Test Your Understanding
Javier and Carmen share some money in the ratio 5 : 6. When each are given £4 more, the ratio of the money they have is now 6 : 7. How much money did each initially have? ?
Using an algebraic approach:
Amounts Javier and Carmen have before:
𝟓𝒙 𝟔𝒙
Amounts after:
𝟓𝒙+𝟒 𝟔𝒙+𝟒
Therefore:
𝟔:𝟕 = 𝟓𝒙+𝟒 :𝟔𝒙+𝟒
𝟓𝒙+𝟒 𝟔 = 𝟔𝒙+𝟒 𝟕
𝟕 𝟓𝒙+𝟒 =𝟔 𝟔𝒙+𝟒 𝟑𝟓𝒙+𝟐𝟖=𝟑𝟔𝒙+𝟐𝟒 𝒙=𝟒
Originally amounts are therefore 𝟓𝒙→𝟐𝟎 and 𝟔𝒙→𝟐𝟒

27. Exercise 3 ? ? ? ? ? (on provided worksheet)
The ratio of children to adults on a school trip is initially 10 : 1. This doesn’t meet government regulations, so 5 more children and 5 more adults join the trip so that the ratio is now 9 : 1. How many children are there now?
405
[JMO 2015 A4] My fruit basket contains apples and oranges. The ratio of apples to oranges in the basket is 3 : 8. When I remove one apple the ratio changes to 1 : 3. How many oranges are in the basket? Solution: 24
The ratio of red to blue counters is 2 : 5 and there are 40 blue counters. When I remove some red counters the ratio is now 1 : 4. How many red counters did I remove?
16 – 10 = 6 removed
The ratio of green to yellow beads is 4 : 5, and there are initially 27 beads. I add some yellow beads and the ratio is now 1 : 3. How many yellow beads did I add?
36 – 15 = 21 beads
The ratio of boys to girls is 1 : 2. Two boys and two girls enter the room and the ratio is now 5 : 9. How many boys and girls were there originally?
8 boys and 16 girls 4 1 ? ? 2 5 ? 3 ? ?

28. Exercise 3 ? ? (on provided worksheet)
[JMO 2004 A8] A large pan contains a mixture of oil and water. After 2 litres of water are added to the original contents of the pan, the ratio of oil to water is 1:2. However, when 2 litres of oil are added to the new mixture, the ratio becomes 2:3. Find the original ratio of oil to water in the pan Solution: 3:5
[STMC Final 2007/08 Q1] Anne, Becky and Charlotte had sums of money in the ratio 7:6:5. One of them gave £9 to one of the others and this changed the ratio (in the same order of names) to 6:5:4. The total sum of money remained the same; what was it? Solution: £810 6 N ? ?