Percentages • Topic 5 of 5

Percentage Applications (CAT)

This is where percentages pay off in CAT: the "product is constant" family and exam-style word problems. When two quantities multiply to a fixed product (price × consumption = expenditure, speed × time = distance, length × breadth = area), a rise of p% in one forces a fall of [p/(100+p)]×100% in the other to keep the product fixed. Example: if price rises 25%, consumption must fall 25/125 = 20% to keep spending the same. The other staples are: passing marks ("scored x%, failed by m marks, pass mark is p%"), election problems (winner’s margin as a % of votes), and expenditure/income splits. The trick is to assign the unknown whole a value of 100 (or the LCM of the denominators) so every percentage becomes a clean number.

✅ Solved examples

1. The price of sugar rises 25%. By what percent must a family cut consumption to keep its sugar bill unchanged?
Cut = [25/(100+25)] × 100 = 25/125 × 100 = 20%.
2. A student scores 30% and fails by 30 marks; passing is 40%. Find the maximum marks.
The 10% gap (40−30) equals 30 marks ⇒ 1% = 3 ⇒ 100% = 300 marks.
3. In an election between two candidates, the winner gets 60% of votes and wins by 1,200 votes. Total valid votes?
Margin = 60% − 40% = 20% = 1200 ⇒ 1% = 60 ⇒ total = 6,000 votes.
4. A man spends 80% of his income. His income rises 20% and expenditure 10%. New savings as % of new income?
Take income 100: spends 80, saves 20. New income 120; new spend = 80×1.1 = 88; new savings = 32 ⇒ 32/120 ≈ 26.67%.

✏️ Practice — try these, take hints as needed

1. Petrol price rises 20%. % cut in use to keep the bill same?
p/(100+p).
20/120.
×100.
16⅔%
2. A scores 25%, fails by 40 marks; pass is 45%. Max marks?
Gap = 45−25 = 20%.
20% = 40 marks.
1% = 2.
200
3. Winner gets 55% and wins by 900 votes. Total votes?
Margin = 55−45 = 10%.
10% = 900.
1% = 90.
9,000
4. Price of an item falls 20%. % rise in consumption to keep spend constant?
p/(100−p) for a fall.
20/80.
×100.
25%
5. A saves 25% of income. Income +20%, expenditure +20%. New savings rate?
Everything scales by 1.2.
Ratio unchanged.
Savings % is the same.
25%

📝 Topic test — 8 questions

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