Arithmetic » Ratios

1. What is a ratio?
2. Calculations with ratios
3. Using a ratio table
4. Cross multiplication

1. What is a ratio?

A ratio between two numbers indicates how one quantity relates to another. When two quantities relate via a ratio they have a directly proportional relationship.

Examples
The ratio between the amount of syrup and water is 1 to 5.
The ratio between the number of cookies you can bake and the amount of flour you need is 2 to 35.
The ratio between the price in eurocents and grams of cheese is 41 to 50.

If you make one of the numbers in a ratio three times larger, you have to make the other number in the ratio also three times larger. In the example of the syrup you will get 3 to 15.

Other notations

1 to 5 = 1 : 5 = 1 / 5.
(You always say 'the ratio is 1 to 5')

This shows that fractions are also ratios.
A ratio that is often used, is the percentage.
The ratio 1 to 5 is the same as 20 to 100 and you can write this as 20%.

But watch out:

Percentages normally are used to compare to the total!
In the example of the syrup and water of 1 to 5 the 5 corresponds to 'water' and not the total! So, here the amount of syrup is 20% of the amount of water! If you would look at the total, there are 1 + 5 = 6 parts. Therefore 16 is syrup and 56 is water. The corresponding percentages are 1 : 6 × 100% ≈ 16,7% syrup and 5 : 6 × 100% ≈ 83,3% water.

2. Calculations with ratios

Example 1
The number of cookies you can bake and the amount of flower you need is 2 to 35.
How many grams of flour do you need for 15 cookies?

Answer:
First calculate the number of grams per cookie and then for 15 cookies.
35 : 2 × 15 = 262.5 grams of flour.

Example 2
The ratio between the price in eurocents and grams of cheese is 41 to 50.
How many euros is a piece of cheese weighing 875 grams.

Answer:
Calculate first the price per gram and then for 875 grams.
41 : 50 × 875 = 717.5 eurocents thus 7.18 euros.

Example 3
Calculate 35% of 85 euros.

Answer:
35 out of 100 of 85.
35 : 100 × 85 = 29.75 euros.

Example 4
How many per cent reduction do you get on a pair of jeans when you have to pay 19.20 euros less then the normal price, which is 80 euros.

Answer:
The ratio is 19.20 to 80. You have to calculate the reduction per euro and then per 100 euros.
19.20 : 80 × 100 = 24
The reduction was 24%.

3. Using a ratio table

In a ratio table you can always multiply the numbers in the top row with a certain number to get the numbers in the bottom row.
If you go for example one column to the right you have to multiply or divide both the top row and the bottom row with the same number.

Example

		×2,5		×2
`grams`	200		500		1000	× 0.015
`euros`	3		7.50		15
		×2.5		×2

Because every number in the top row can be multiplied with 0.015 to get the numbers in the bottom row, this is a ratio table.

Example 1
If 1 kg costs 3.25 euros, how much is 450 grams?

Answer:
Make a ratio table with 1000 grams and 3.25 euros.
Then calculate towards 450 grams.

		:1000		×450
`grams`	1000		1		450	The answer is (rounded off) 1.46 euros.
`euros`	3.25		...		1.46
		:1000		×450

The intermediate answer at … is not written down. This is to prevent mistakes with rounding off in the middle of your calculation

Example 2
When you need 3 eggs for 450 grams of cookies, how many cookies can you bake with 5 eggs?

Answer:
Make a table with the ratio 450 grams and 3 eggs. Then calculate towards 5 eggs.

		:3		×5
`grams`	450		...		750	The answer is 750 grams of cookies.
`eggs`	3		1		5
		:3		×5

Example 3
How many cars were driving too fast when 21% of the 7500 passing cars got a speeding ticket?

Answer:
Make a table with number and percentage and fill in that 7500 is 100%. Then calculate towards 21%.

		:100		×21
`number`	7500		...		1575	The answer is 1575 cars.
`percentage`	100		1		21
		:100		×21

Example 4
If you get 369 euros reduction on a washing machine that normally costs 1230 euros, how many per cent is you reduction?

Answer:
Make a ratio table with number and percentage and fill in that 1230 euros is 100%. Then calculate towards 369 euros.

		:1230		×369
`number`	1230		1		369	The answer is 30%.
`percentage`	100		...		30
		:1230		×369

4. Cross multiplication

In a ratio table you can use cross multiplication.
When you want to calculate an unknown value, you can multiply the number next to it with the number above it and divide it with the number diagonally above it.
Knowing this, you can work a lot faster when using ratio tables.

How does it work?

The following rule applies to the table below: A × D = B × C

A	C
B	D

This gives:
A = B × C : D
B = A × D : C
C = A × D : B
D = B × C : A

Examples

35	150
7	D

D = 7 × 150 : 35 = 30

28	C
100	14

C = 14 × 28 : 100 = 3,92

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