Ratios
Wackford Squeers
Ratios Simplify ratios Work out ratios of quantities Suppose you have 5 sweets and you want to give them to your brother and sister according to their ages You might decide to give one 2 sweets and the other 3 sweets
Now suppose you had 10 sweets and wanted to keep the shares the same. You would again need to give one 2 sweets and the other another 3 sweets
And so on ‌‌
2
What you have been doing is sharing the sweets out according to a ‘rule’ The rule in the example was 2:3 and is called the
Ratio
If we look at what we did and the numbers we see that: We had 5 sweets to share One got 3 sweets and the other 2 sweets In the second example: We had 10 sweets to share One got 6 sweets and the other 4 sweets If we had had 30 sweets we would have had to share them: 18 sweets to one child 12 sweets to the other child And we notice that ALWAYS the number of sweets we have shared out equals the number of sweets we started with For example 18 + 12 = 30 3+2=5 6 + 4 = 10 We now have the beginnings of a method for working out ratios: 1. Add the shares together to get a total number of shares 2. Divide this total into the given quantity to see how much one share is worth 3. Multiply each part of the sharing rules by the value of the share Let’s see the system in practice
3
Example 1 Share 24 sweets in the ratio of Here we have
3:5
3 + 5 shares = 8 shares
We have 24 sweets to share between 8 shares So one share is worth 24 ÷ 8 =
3
Now we have the value of one share we can work out the shares One person is to get
3 shares = 3 x 3 = 9 sweets
The other person is to get
5 x 3 = 15 sweets
CHECK: 9 + 15 = 24
Example 2 Share £21 in the ratio of
3:4
Total number of shares = 3 + 4 = 7 Each shares is worth 21 ÷ 7 =
3
3 shares are worth
3 x 3 = £9
4 shares are worth
4 x 3 = £12
So the share-out is £9 and £12
Exercises
4
Exercises 1 Calculate the following ratios: 1 2 3 4 5 6
24 in the ratio of 3:5 15 in the ratio of 2:3 21 in the ratio of 2:5 63 in the ratio of 5:4 120 in the ratio of 7:5 300 in the ratio of 2:8
7 8 9 10 11 12
45 in the ratio of 4:5 72 in the ratio of 3:5 150 in the ratio of 1:2:3 270 in the ratio of 2:3:4 360 in the ratio of 3:4:5 63 in the ratio of 2:3:4
Simplifying Ratios Sometimes you are given a ratio which can be made more simple, before it is used For example, the ratio:
16:24 Is perfectly OK – just a little difficult to work with However, if we were to divide both sides by 8 we would get
2:3 A much simpler ratio to work with The rules are quite straightforward: Divide BOTH SIDES by the same number to reduce the ratio to its simplest form (i.e. when it can’t be cancelled down any more)
5
Exercises 2 Simplify the following ratios 1 2 3 4 5 6 7 8 9 10
4:6 9:12 16:24 3:15 8:20 5:15 6:30 9:21 8:18 15:50
11 12 13 14 15 16 17 18 19 20
14:56 8:20 35:80 25:40 15:40 24:42 63:36 49:105:21 18:30:42 27:45:81
Ratios with different units Sometimes you have a ratio where the units in the two (or more) parts are in different units – e.g. cm and m This is not a problem, but you do need to make sure that all parts of the ratio are in the same unit
Example Simplify 5cm:2m Here we have cm and metres We need to change one of them to the same unit as the other. It doesn’t make any difference which one, but, in this case, it is probably easier to change the metres to centimetres Since there are 100cm in a metre the 2m becomes 200cm And we have a ratio in the same unit:
5:200 Which we can cancel down to
1:40
6
Exercises 3 – More Complex Simplify the following (HINT: Look at the units!) 1 2 3 4 5
10cm:50cm 10cm:5m 15mm:10cm 40m:50cm 27cm:81m
6 7 8 9 10
5km:500m 5cm:1km 35cm:14km 2km:40m 1km:2.5m
Exercises 1 – Answers 1 2 3 4 5 6
9 and 15 6 and 9 6 and 15 35 and 28 70 an 50 60 and 240
7 8 9 10 11 12
20 27 25 60 90 14
and 25 and 45 50 75 90 120 120 150 21 28
11 12 13 14 15 16 17 18 19 20
1:4 2:5 7:16 5:8 3:8 4:7 7:4 7:15:3 3:5:7 3:5:9
6 7 8 9 10
10:1 1:20000 1:4000 50:1 400:1
Exercises 2 – Answers 1 2 3 4 5 6 7 8 9 10
2:3 3:4 2:3 1:5 2:5 1:3 1:5 3:7 4:9 3:10
Exercises 3 – Answers 1 2 3 4 5
1:5 1:50 3:20 80:1 1:300
7