EDEXCEL GCSE HIGHER MATHS - What are Ratios, Direct and Inverse Proportion

EDEXCEL GCSE Higher Maths: Ratios, Direct & Inverse Proportion

Ratios

What is a ratio?

A ratio compares two or more quantities. It tells us how much of one quantity there is compared to another. We can write ratios in a few different ways:

• Using a colon: 3 : 4
• As a fraction: 3/4
• Using the word "to": 3 to 4

Simplifying ratios:

Ratios can be simplified just like fractions. Divide both parts of the ratio by their highest common factor.

Example:

Simplify the ratio 12 : 18.

• The highest common factor of 12 and 18 is 6.
• Divide both parts of the ratio by 6: 12 ÷ 6 : 18 ÷ 6 = 2 : 3

Sharing in a ratio:

Ratios are often used to share amounts. To share an amount in a given ratio, follow these steps:

1. Find the total number of parts: Add the numbers in the ratio.
2. Divide the amount by the total number of parts: This gives you the value of one part.
3. Multiply the value of one part by the ratio numbers: This gives you the amount each person receives.

Example:

Two friends share £60 in the ratio 2 : 3. How much does each friend get?

1. Total parts: 2 + 3 = 5
2. Value of one part: £60 ÷ 5 = £12
3. Amount for each friend:
4. Friend 1: £12 × 2 = £24
5. Friend 2: £12 × 3 = £36

Direct Proportion

What is direct proportion?

Two quantities are in direct proportion if they increase or decrease at the same rate. This means that if one quantity doubles, the other also doubles.

Recognizing direct proportion:

• The ratio of the quantities remains constant.
• The graph of the relationship is a straight line passing through the origin.

Finding the constant of proportionality:

The constant of proportionality is the number that relates the two quantities in direct proportion. To find it, divide one quantity by the corresponding value of the other quantity.

Example:

The number of cakes baked is directly proportional to the amount of flour used. If 3 cakes require 250g of flour, how much flour is needed to bake 12 cakes?

1. Find the constant of proportionality: 250g ÷ 3 = 83.33g per cake
2. Calculate the flour needed for 12 cakes: 83.33g/cake × 12 cakes = 1000g

Inverse Proportion

What is inverse proportion?

Two quantities are in inverse proportion if one quantity increases as the other decreases at a proportional rate. This means that if one quantity doubles, the other halves.

Recognizing inverse proportion:

• The product of the two quantities remains constant.
• The graph of the relationship is a curve that approaches the axes but never touches them.

Finding the constant of proportionality:

The constant of proportionality is the product of the two quantities in inverse proportion.

Example:

The time taken to travel a certain distance is inversely proportional to the speed. If it takes 3 hours to travel at a speed of 60km/h, how long will it take to travel at a speed of 90km/h?

1. Find the constant of proportionality: 3 hours × 60km/h = 180
2. Calculate the time taken at 90km/h: 180 ÷ 90km/h = 2 hours

Key Points to Remember

• Ratios compare quantities.
• Direct proportion means quantities increase or decrease at the same rate.
• Inverse proportion means one quantity increases as the other decreases at a proportional rate.
• The constant of proportionality is a key value in both direct and inverse proportion.

By understanding ratios, direct and inverse proportion, you'll be well-equipped to tackle a range of problems in your EDEXCEL GCSE Higher Maths exam.