QUANT SHORTCUT | 23 OCTOBER 2026
Two numbers are in the ratio 3 : 5. If 10 is subtracted from each, the new ratio becomes 4 : 7. Find the numbers. The answer is 90 and 150, and Amit Sir reaches it without writing a single equation, using one idea: when you add or subtract the same number from both, their difference does not change. This page explains the idea, walks through his method step by step, checks it with algebra, and gives you worked examples and 10 practice questions.
The trap: multiplying the ratio by 10
A common slip is to see “10” in the question and simply multiply the ratio by it, answering 30 and 50. Check it: 30 − 10 = 20 and 50 − 10 = 40, and 20 : 40 is 1 : 2, not 4 : 7. The 10 is the amount taken away, not the size of one part, so the guess fails. Always test an answer against the second condition before you mark it.
The key idea: the difference stays the same
If two numbers are A and B, then (A − 10) and (B − 10) have the same difference as A and B, because the 10 cancels: (B − 10) − (A − 10) = B − A. The same is true if you add any number to both. So the actual gap between the two numbers is the one thing that both ratios share. Amit Sir’s trick is to make the two ratios agree on that gap, so that both are written in the same units.
Amit Sir’s method, step by step
- Find each difference: in 3 : 5 the difference is 5 − 3 = 2; in 4 : 7 it is 7 − 4 = 3.
- Make the differences equal: the LCM of 2 and 3 is 6, so multiply 3 : 5 by 3 and 4 : 7 by 2. You get 9 : 15 and 8 : 14. Both now have a difference of 6, so one unit means the same amount in both.
- Read the change: the first number went from 9 units to 8 units, so it lost exactly 1 unit.
- That unit is the 10: 10 was taken away, so 1 unit = 10.
- Scale back: the original numbers are 9 units and 15 units, so they are 9 × 10 = 90 and 15 × 10 = 150.
Check it with algebra
Let the numbers be 3x and 5x. Then (3x − 10) : (5x − 10) = 4 : 7, so 7(3x − 10) = 4(5x − 10), which gives 21x − 70 = 20x − 40 and x = 30. The numbers are 3 × 30 = 90 and 5 × 30 = 150, the same as the trick. Final check: 90 − 10 = 80 and 150 − 10 = 140, and 80 : 140 = 4 : 7.
More worked examples
- Adding instead of subtracting: two numbers are 2 : 5, and adding 8 to each makes 1 : 2. Differences are 3 and 1, so write 1 : 2 as 3 : 6. The first number went from 2 units to 3 units, so 1 unit = 8, and the numbers are 16 and 40 (check: 24 : 48 = 1 : 2).
- Differences already equal: 4 : 5 becomes 5 : 6 when 6 is added to both. Both differences are 1, so 4 → 5 is 1 unit = 6, and the numbers are 24 and 30 (check: 30 : 36 = 5 : 6).
- Ages: a father and son’s ages are 7 : 2; ten years later they will be 9 : 4. Differences 5 and 5 already match, so 7 → 9 is 2 units = 10 years, 1 unit = 5, and the ages are 35 and 10 (check: 45 : 20 = 9 : 4).
If a : b becomes c : d after adding (or subtracting) k to both, scale both ratios so that b − a and d − c are equal; then the change in one term, in units, equals k, which gives the value of one unit.
- Age questions in Quantitative Techniques are this exact pattern: the age gap between two people never changes.
- Mixture and “add the same amount to both” questions in data-based sets use the same idea.
- The trick is fastest when the differences are small; scale them to their LCM, not to their product.
- Always verify with the second ratio: one subtraction and one division catch most slips.
| Fact | Detail |
|---|---|
| Question | 3 : 5, both − 10 → 4 : 7, find the numbers |
| Differences | 5 − 3 = 2 and 7 − 4 = 3 |
| Equalise (LCM 6) | 3 : 5 × 3 = 9 : 15; 4 : 7 × 2 = 8 : 14 |
| Change in units | 9 → 8 = 1 unit = 10 |
| Answer | 9 × 10 = 90 and 15 × 10 = 150 |
| Check | 80 : 140 = 4 : 7 |
| Wrong answer to avoid | 30 and 50 (multiplying the ratio by the 10) |
Add or take away the same amount and the gap stays put. Match the gaps first; the change you see is the number in the question.
Sources
Amit Sir’s CLAT Gurukul class recording (the 2-Second Trick series, episode 6); the algebra above is standard ratio arithmetic.
Free in the CLAT Gurukul app
These notes (as a PDF lesson) and the 10-question CBT are in the app under Current Affairs → 23 Oct 2026. Get the app on Google Play or open app.clatgurukul.com.
Practice Quiz — 10 CLAT-Style Questions
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