UPSC Prelims 2009 Q105: Science & Technology – Quantitative Aptitude
There are 240 balls and n number of boxes B1, B2, B3, … , Bn. The balls are to be placed in
the boxes such that B1 should contain 4 balls more than B2, B2 should contain 4 balls more than
B3, and so on.
Which one of the following cannot be the possible value of n?
- (a) 4
- (b) 5
- (c) 6
- (d) 7
Answer: (d) 7
Explanation
Which value of n cannot distribute 240 balls under the given rule
Boxes B1 to Bn hold a decreasing sequence of balls, each 4 more than the next, summing to 240.
| Step | Reasoning |
|---|---|
| Setup | Let the last box Bn hold x balls; the sequence is x, x+4, x+8, ..., up to n terms |
| Sum formula | Sum = n·x + 4·(0+1+...+(n−1)) = n·x + 2n(n−1) = 240 |
| Test n=4 | 4x + 24 = 240 → x = 54 (valid, positive integer) |
| Test n=5 | 5x + 40 = 240 → x = 40 (valid) |
| Test n=6 | 6x + 60 = 240 → x = 30 (valid) |
| Test n=7 | 7x + 84 = 240 → x = 156/7 = 22.28... (NOT a whole number — invalid) |
Why n=7 fails:
| Point | Detail |
|---|---|
| Box contents must be whole numbers | n=7 gives a non-integer value for the smallest box |
| n=4, 5, 6 | All give valid whole-number solutions |
Answer: (d) 7.
Why each option is right or wrong
- (a) Incorrect. n=4 gives a valid whole-number solution (x=54), so this IS a possible value.
- (b) Incorrect. n=5 gives a valid whole-number solution (x=40), so this IS a possible value.
- (c) Incorrect. n=6 gives a valid whole-number solution (x=30), so this IS a possible value.
- (d) CORRECT. n=7 gives a non-integer result (22.28...), making it the value of n that CANNOT work.
Key point
Testing each value of n in the equation n·x + 2n(n−1) = 240, only n=7 fails to produce a whole-number box count — making it the impossible value.
Remember
240 balls, boxes decreasing by 4 each: n=7 is the value that does NOT give a whole-number solution.
Common mistakes
- Forgetting that box contents must be whole numbers, and not checking for non-integer results.
- Making an arithmetic error while setting up or solving the sum equation for each candidate n.
- Testing only some values of n instead of systematically checking all four options.
Exam tip
For 'decreasing sequence totaling X' problems, test each candidate n in the sum formula and check whether the result is a valid whole number.
Sources
- NCERT Class X Mathematics — Arithmetic Progressions
- NCERT Class XI Mathematics — Sequences and Series