Introduction to Ratios

 

What is a Ratio?

A ratio is a way of comparing one part of a whole to another. It tells us the relative sizes of two or more quantities.

Ratios are used in real life for:

• Mixing ingredients in a recipe
• Splitting money between people
• Comparing distances on a map

A ratio compares one part to another part or one part to the whole.

 

What Do Ratios Look Like?

Ratios involve two or more numbers separated by a colon (:).

Examples:

• $2:5$
• $3:1$
• $4:2:3$

Key Rule: The order in a ratio matters For example, in a recipe with flour and butter in the ratio 2:1,

• Flour is associated with 2
• Butter is associated with 1

The numbers in a ratio tell us how much of each quantity is present compared to the total.

 

Understanding Parts of a Ratio

In the ratio $4:3$:

• The first quantity makes up 4 parts
• The second quantity makes up 3 parts
• The total number of parts is 4 + 3 = 7

In the ratio $2:5:3$:

• First quantity = 2 parts
• Second quantity = 5 parts
• Third quantity = 3 parts
• Total parts = 2 + 5 + 3 = 10

 

Example

A pot of money is shared between three friends: Dave, John, and Mary.

Dave receives $\pounds450$, John receives $\pounds 200$, and Mary receives $\pounds 350$.

(a) Find the total amount of money in the pot. $$450 + 200 + 350 = 1000$$

Total money = $\pounds 1000$

(b) Write the ratio of money received by Dave, John, and Mary.

• Dave gets $\pounds 450$, John gets $\pounds 200$, and Mary gets $\pounds 350$
• So the ratio is: $$450:200:350$$

(c) Find the fraction of the total money that Mary receives.

• Mary's share = $\pounds 350$
• Total money = $\pounds 1000$  $$\frac{350}{1000}$$

 

Equivalent Ratios

Two ratios are equivalent if they represent the same proportion.

For example:

The ratio $5:10$ is equivalent to $20:40$ because: $$5 \times 4 = 20, \quad 10 \times 4 = 40$$

 

Real-Life Example: Scaling a Recipe

A cake recipe has flour and butter in the ratio 3:2.

Using just 3g of flour and 2g of butter would make a tiny cake. A more reasonable equivalent ratio would be 300:200 (300g flour, 200g butter).

 

How to Find an Equivalent Ratio

Multiply or divide all parts of the ratio by the same number.

Example:

Find an equivalent ratio to $2:3:7$ by multiplying each part by 4.

$$(2 \times 4) : (3 \times 4) : (7 \times 4) = 8:12:28$$

Key Rule: Equivalent ratios keep the same relative proportion but use different numbers.

Common Mistake: $1:4$ is NOT the same as $\frac{1}{4}$

 

 

 

 

 

Simplifying Ratios

A ratio is in simplest form when:

• All numbers in the ratio are whole numbers
• There are no common factors between the numbers

Example:

The ratio 45:30 simplifies to 3:2 because:

$$45 \div 15 = 3, \quad 30 \div 15 = 2$$

 

How to Simplify a Ratio

1. Find the Highest Common Factor (HCF) of all numbers.
2. Divide each number in the ratio by the HCF.

Example: Simplify 30:66:12

• HCF of 30, 66, and 12 is 6
• Divide each number by 6:

$$(30 \div 6) : (66 \div 6) : (12 \div 6) = 5:11:2$$

Final Answer: 5:11:2