Ratios

… Part-to-total ratios.

This is where an amount is shared between certain ratios. For example, if Alice and Ben share £400 in the ratio 5:3, how much does Ben get?

Firstly, find the total number of ‘parts’ = 5 + 3 = 8

Next, find out how much each part is worth = £400/8 = £50

And finally multiply by the ratios:

Alice : Ben

5 ×50 : 3 ×50

£250 : £150

… Part-to-part ratios.

In these questions, you will need to find scaling factors to move between ratios. For example, chocolate milkshake is made by mixing milk and ice cream in the ratio 2:9. How much milkshake is made if 801ml of ice cream is used?

Milk : Ice cream

2 : 9

x : 801

To move from 9 to 801, the scaling factor is ÷9 ×801

Apply the same scaling factor to the milk side of the ratio:

x = 2 ÷ 9 × 801 = 178ml of milk is needed

Therefore, total volume of milkshake made = 178 + 801 = 979ml

… Converting currencies using an exchange rate ‘ratios’ method:

In these questions, you will need to find scaling factors to move between ratios. For example, if the current exchange rate is £1 = $1.43, then how much does a $120 CD set cost in pounds £?

£ : $

1 : 1.43

: 1

x : 120

To move from $1.43 to $1, the scaling factor is ÷1.43

To move from $1 to $120, the scaling factor is ×120

Apply the same scaling factors to the £ side of the ratio:

x = 1 ÷ 1.43 × 120 = £83.92

Therefore, $120 = £83.92

… When comparisons are made within ratios, it can be a useful technique to multiply by an ‘x’. For example:

A : B

3 : 8

3x : 8x

If, for example, A gets £30 more than B, then:

3x + 30 = 8x … Which can then be solved to find x

… Ratios can involve different units, but each column must use the SAME units:

Compost : Price

25 litres : £8

100 litres : £32

… Sometimes you may need to generate equations from ratios. For example,

if the ratio of c : d is 5 : 3, then we can also write this in fraction form:

c/d = 5/3

… If 3m = 12p, what is the ratio of m : p in its lowest form?

… Ratios may need to be combined by scaling them up until you get a ‘match’, for example:

The ratio A : B = 3 : 4

The ratio B : C = 5 : 6

So

A : B B : C

3 : 4 5 : 6 Now multiply the first ratio by 5 and the second by 4:

15:20 20:24 So we have matched the Bs, and so:

A : B : C = 15 : 20 : 24

… If the ratio of children to adults at a party is 5:2, we could write this ratio as:

C : A

5n : 2n … Where n is a multiplying factor which give us the ACTUAL numbers of people

If 6 more children and 3 more adults arrive, we could then write the new ratio as:

5n + 6 : 2n + 3

Ratios and coordinate geometry

… Sometimes you may need to find the coordinate of a point on a line according to some ratio. For example, if points A, B and C lie on a straight line where

A(-3,-5)

B(1 , 2)

AB : BC = 2 : 3

Find the coordinates of point C

First draw out a little grid showing the ratio and coordinates:

x y

↑ A -3 -4

2 parts

↓ B 1 2

↑

3 parts

↓ C

Step 1) x-coordinates

First look at the x jump from A to B = +4 spaces = 2 parts

So 2 spaces = 1 part

Now look at the x jump from B to C. We need 3 parts = 3 × 2 spaces = 6 spaces

1 + 6 spaces = 7

And this is the x coordinate of point C

Step 2) y-coordinates

First look at the y jump from A to B = +6 spaces = 2 parts

So 3 spaces = 1 part

Now look at the y jump from B to C. We need 3 parts = 3 × 3 spaces = 9 spaces

2 + 9 spaces = 11

And this is the y coordinate of point C

So point C is at (7,11)