Mastering Ratio and Proportion in GCSE Maths

Introduction to Ratios and Proportions

Understanding ratios and proportions is a crucial aspect of GCSE Mathematics. These topics are widely applicable in various real-life situations and form the foundation for more advanced mathematical concepts.

Ratios

A ratio compares two or more quantities of the same kind. It expresses the relative sizes of these quantities, without specifying the units. Ratios can be written using the colon notation (e.g., 2:3) or as a fraction (e.g., 2/3).

Example: Simplifying Ratios

Problem: Simplify the ratio 12:18.

Solution:

• Divide both terms by their highest common factor (HCF), which is 6.
• 12/6 = 2, 18/6 = 3
• The simplified ratio is 2:3.

Dividing Quantities in a Given Ratio

Ratios can be used to divide quantities in a given proportion. This is particularly useful in scenarios involving sharing or distribution.

Example: Dividing Quantities in a Ratio

Problem: Sam and Alex share $180 in the ratio 3:2. How much does each person receive?

Solution:

• The total number of parts is 3 + 2 = 5.
• Sam's share is (3/5) × $180 = $108.
• Alex's share is (2/5) × $180 = $72.

Direct Proportion

Two quantities are in direct proportion if they increase or decrease at the same rate. This relationship can be represented algebraically as y = kx, where k is the constant of proportionality.

Example: Direct Proportion

Problem: The cost of printing posters is directly proportional to the number of posters. If 10 posters cost $25, find the cost of 30 posters.

Solution:

• Let x be the number of posters and y be the cost.
• We know that y = kx and when x = 10, y = 25.
• Substituting, we get 25 = 10k, so k = 2.5.
• For 30 posters, y = 2.5 × 30 = $75.

Inverse Proportion

Two quantities are in inverse proportion if one increases as the other decreases, and vice versa. This relationship can be represented algebraically as y = k/x, where k is the constant of proportionality.

Example: Inverse Proportion

Problem: The time taken to complete a job is inversely proportional to the number of workers. If 6 workers can complete the job in 8 days, how many days will it take for 12 workers to complete the same job?

Solution:

• Let x be the number of workers and y be the time taken.
• We know that y = k/x and when x = 6, y = 8.
• Substituting, we get 8 = k/6, so k = 48.
• For 12 workers, y = 48/12 = 4 days.

Real-Life Applications and Scale Factors

Ratios and proportions have numerous applications in various fields, including architecture, engineering, economics, and more. Understanding scale factors is crucial in interpreting and creating scale drawings, models, and maps.

In the AQA GCSE Mathematics specification, students are expected to solve problems involving compound measures, such as speed, density, and currency conversions, using ratios and proportions.