Ratio

A ratio is an expression used to compare two or more quantities relative to each other.

The quantities are separated by a colon (:).

Example

There are two cats and one dog.

The ratio of cats to dogs is:

The ratio of dogs to cats is:

This means there are twice as many cats as dogs, or half as many dogs as cats.

Example

There are six cats and three dogs.

There are still twice as many cats as dogs.

Dividing through by the HCF reduces a ratio to its simplest form.

Here, the HCF of 6 and 3 is 3, so divide both sides by 3.

\[ 6 : 3 \] \[ = \frac{6 \div 3}{3 \div 3} \] \[ = 2 : 1 \]

The ratio 6 : 3 reduces to 2 : 1 when expressed in simplest form.

Multiplying the simplest form gives equivalent ratios:

The ratio 8 : 4 is equivalent to 2 : 1, 4 : 2, 200 : 100, etc.

How can this be used?

Example

Given the ratio of boys to girls as 2 : 3, how many girls will there be if there are 10 boys?

Notice that the original number of boys has been multiplied by 5.

This means the original number of girls must also be multiplied by 5.

So there are 15 girls.

Example

Given the ratio of girls to boys as 3 : 2, how many boys will there be if there are 12 girls?

The original number of girls (3 parts) has been multiplied to reach 12.

\[ 3 \to 12 \quad \text{(multiplied by 4)} \]

So the number of boys (2 parts) must also be multiplied by 4.

\[ 2 \times 4 = 8 \]

There are 8 boys.

Notice that the term containing both known quantities (girls and 12) goes on the left.

Other Methods

Ratios are sometimes written as ratio equations.

\[ 3 : 2 \; :: \; 12 : x \]

In Vedic maths , we use the idea that:

The product of the means equals the product of the extremes.

\[ a : b \; :: \; c : d \quad \Rightarrow \quad ad = bc \]

Examples

Find the value of x

3 : 2 :: 12 : x

\[ 3 : 2 \; :: \; 12 : x \] \[ 3x = 2 \times 12 \] \[ 3x = 24 \] \[ x = 8 \]

Find the value of x

5 : x :: 35 : 9

\[ 5 : x \; :: \; 35 : 9 \] \[ 5 \times 9 = 35x \] \[ 45 = 35x \] \[ x = \frac{45}{35} \] \[ x = \frac{9}{7} \]

Example

There are two cats and one dog.

The proportion of cats is: \[ \frac{2}{3} \]

The proportion of dogs is: \[ \frac{1}{3} \]

Two thirds of the animals are cats. One third of the animals are dogs.

Example

Divide 45 kg in the ratio 3 : 2

\[ \text{Total parts} = 3 + 2 = 5 \] \[ \text{Value of each part} = \frac{45}{5} = 9 \] \[ 3 \text{ parts} = 3 \times 9 = 27 \text{ kg} \] \[ 2 \text{ parts} = 2 \times 9 = 18 \text{ kg} \]

So the amounts are 27 kg and 18 kg.

(Check: 27 + 18 = 45)

Example

A box of 84 chocolates is shared amongst Tom, Ryan and Ashleigh in the ratio 3 : 4 : 5. How many chocolates does each person receive?

\[ \text{Total parts} = 3 + 4 + 5 = 12 \] \[ \text{Value of each part} = \frac{84}{12} = 7 \] \[ \text{Tom} = 3 \times 7 = 21 \] \[ \text{Ryan} = 4 \times 7 = 28 \] \[ \text{Ashleigh} = 5 \times 7 = 35 \]

So they receive 21, 28 and 35 chocolates.

Direct Proportion Indirect Proportion Variation