IGCSE Maths / Ratios and percentages
Changing ratios revision notes
Edexcel International GCSE Maths (4MA1), chapter 2: Ratios and percentages, specification point 1.7e. Free lesson.
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When a ratio changes, use algebra to represent the starting amounts and form an equation from the new ratio. If one person's share is given, divide by their number of parts to find one part.
Find one part from a person's share
Each number in a ratio tells you how many equal parts a person gets. If you know one person's share, divide it by that person's parts, not by the total parts. Kofi and Lena share money in the ratio 4 : 7. Lena's £84 is her 7 parts, so one part is £84 ÷ 7 = £12. Kofi has 4 parts, so he gets 4 × £12 = £48. Check that Lena's 7 parts give £84.
Use x when the ratio changes
When something is taken away or given to someone else, let one part be x. Write both starting amounts using x, then use what happens to form an equation. Sam and Tia start in the ratio 5 : 3, so they have 5x and 3x sweets. Sam eats 12, leaving 5x − 12. They now have the same number, so set this equal to Tia's 3x.
Build an equation from the new ratio
For a transfer, subtract the amount given away from the giver's amount and add it to the receiver's. Then use the new ratio and cross multiply to make an equation. Ali and Ben start in the ratio 7 : 3, so their amounts are 7x and 3x. Ali gives Ben £20. Their new amounts are 7x − 20 and 3x + 20, and the new ratio is 3 : 2.
Use x to find the full amount
The number mentioned in a question is not automatically one part. x represents one part, so use the value you solve for to calculate the person's starting amount. In the sweets example, 12 is how many sweets Sam eats, not the size of one part. Since x = 6, Sam started with 5 × 6 = 30, not 5 × 12 = 60. Mia and Zak start in the ratio 8 : 1. Mia gives Zak 5 stickers, and the new ratio is 3 : 1. The new amounts are 8x − 5 and x + 5. Cross multiply: 8x − 5 = 3(x + 5), so 8x − 5 = 3x + 15 and x = 4. Mia started with 8 × 4 = 32 stickers.