UPSC Prelims 2009 Q99: Science & Technology – Quantitative Aptitude
There are four persons A, B, C, D; and A has some coins. A gave half of the coins to B and 4
more besides. B gave half of the coins to C and 4 more besides. C gave half of the coins to D and 4
more besides. Both B and D end up with same number of coins. How many coins did A have
originally?
- (a) 96
- (b) 84
- (c) 72
- (d) 64
Answer: (c) 72
Explanation
A backward-solving coin-sharing puzzle among four persons
Four persons — A, B, C, D — pass along coins following a repeated "half plus 4" rule, with persons B and D ending up with the same number of coins — we need the number of coins person A had originally.
| Step | Working forward with A's original coins = x |
|---|---|
| Person A gives person B | x/2 + 4 coins; A keeps x/2 − 4 coins |
| Person B gives person C | (x/2+4)/2 + 4 = x/4 + 6 coins; B keeps (x/2+4) − (x/4+6) = x/4 − 2 coins |
| Person C gives person D | (x/4+6)/2 + 4 = x/8 + 7 coins; C keeps (x/4+6) − (x/8+7) = x/8 − 1 coins |
| Person D ends up with | x/8 + 7 coins |
| Setting B's final coins equal to D's final coins | x/4 − 2 = x/8 + 7 |
| Solving | Multiply by 8: 2x − 16 = x + 56 → x = 72 coins |
Answer: (c) 72.
Why each option is right or wrong
- (a) Incorrect. 96 does not satisfy the final equation when checked.
- (b) Incorrect. 84 also fails to satisfy B's and D's final amounts being equal.
- (c) CORRECT. Solving the step-by-step 'half plus 4' equations gives x = 72, verified by B's and D's final equal amounts.
- (d) Incorrect. 64 does not satisfy the equation either.
Key point
Working through each 'half plus 4' transfer step by step and setting B's final coin count equal to D's final coin count gives A's original amount: 72.
Remember
Coin-sharing puzzle (half + 4 rule, B and D end equal): A originally had 72 coins.
Common mistakes
- Making an algebraic error while tracking each person's running coin total through multiple transfer steps.
- Forgetting to apply the 'plus 4' part of the rule at each step, not just the halving.
- Setting up the final equation incorrectly, mismatching which two people's amounts should be equal.
Exam tip
For sequential 'half plus a fixed amount' puzzles, carefully track each person's running total algebraically, then set up the final equality condition precisely as stated.
Sources
- NCERT Class VIII Mathematics — Linear Equations in One Variable
- Arun Sharma — How to Prepare for Quantitative Aptitude for the CAT (Algebra)