UPSC Prelims 2010 Q53: Science & Technology – Quantitative Aptitude
Two numbers X and Y are respectively 20% and 28% less than a third number Z. By what
percentage is the number Y less than the number X?
- (a) 8%
- (b) 9%
- (c) 10%
- (d) 12%
Answer: (c) 10%
Explanation
Percentage comparison: how much less is Y than X?
Both X and Y are defined relative to a third number Z.
| Step | Calculation |
|---|---|
| Assume Z = 100 | For easy percentage math |
| X = 20% less than Z | X = 100 − 20 = 80 |
| Y = 28% less than Z | Y = 100 − 28 = 72 |
| Difference (X − Y) | 80 − 72 = 8 |
| Percentage Y is less than X | (8 / 80) × 100 = 10% |
Why we divide by X, not Z:
| Point | Detail |
|---|---|
| Question asks | 'Y less than X' — so the base for percentage is X, not Z |
| Result | Y is 10% less than X |
Answer: (c) 10%.
Why each option is right or wrong
- (a) Incorrect. 8% incorrectly uses Z (100) as the base instead of X (80).
- (b) Incorrect. 9% does not match the correct division of the difference by X.
- (c) CORRECT. (80−72)/80 × 100 = 10%, using X as the correct base for the comparison.
- (d) Incorrect. 12% does not match the correct calculation for 'Y less than X'.
Key point
With Z=100, X=80 (20% less) and Y=72 (28% less); the percentage Y is less than X = (X−Y)/X × 100 = 8/80 × 100 = 10%.
Remember
X=80, Y=72 (from Z=100): Y is 10% less than X — using X as the base, not Z.
Common mistakes
- Using the wrong base number (Z instead of X) when calculating the percentage difference.
- Confusing 'Y is X% less than Z' problems with 'Y less than X' comparison problems, which need a different base.
- Making an arithmetic slip in the final division step.
Exam tip
'Y less than X' problems: always use X as the base for the percentage calculation, not the original reference number Z.
Sources
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Percentage)
- NCERT Class VIII Mathematics — Comparing Quantities