$M$ men agree to purchase a gift for ₹ $D$. If three men drop out how much more will each have to contribute towards the purchase of the gift?

### Concept & Strategy This is an algebraic fraction problem. The "extra" contribution required is simply the difference between the new per-person cost and the original per-person cost. $$ \text{Extra Contribution} = \text{New Share} - \text{Old Share} $$ ### Step-by-Step Solution * **Calculate the Original Share:** The total cost is $D$ and the original number of men is $M$. * Original Share per man = $\frac{D}{M}$. * **Calculate the New Share:** 3 men drop out, so the new number of men is $M - 3$. The total cost $D$ remains unchanged. * New Share per man = $\frac{D}{M - 3}$. * **Calculate the Difference:** Subtract the old share from the new share to find the increase. * Increase = $\frac{D}{M - 3} - \frac{D}{M}$. * **Simplify the Expression:** Find a common denominator, which is $M(M - 3)$. * Multiply the first fraction by $\frac{M}{M}$: $\frac{DM}{M(M - 3)}$ * Multiply the second fraction by $\frac{M - 3}{M - 3}$: $\frac{D(M - 3)}{M(M - 3)}$ * Combine numerators: $\frac{DM - D(M - 3)}{M(M - 3)}$ * Expand the numerator: $DM - DM + 3D = 3D$. * Expand the denominator: $M^2 - 3M$. * Final expression: $\frac{3D}{M^2 - 3M}$. ### Exam Strategy & Shortcut When variables confuse you under pressure, plug in simple numbers. Let $D = 20$ (Total cost) and $M = 5$ (Total men). Original share = $20 / 5 = \text{₹ } 4$. If 3 drop out, new men = $2$. New share = $20 / 2 = \text{₹ } 10$. Increase = $10 - 4 = \text{₹ } 6$. Now, plug $D=20, M=5$ into the options and look for the one that equals 6: (a) $20 / 2 = 10$ (Incorrect) (b) $100 / 3 = 33.3$ (Incorrect) (c) $5 / 17$ (Incorrect) (d) $(3 \times 20) / (25 - 15) = 60 / 10 = 6$. (Match!) ### Common Pitfall Subtracting the denominators incorrectly or selecting option (a), which is just the *new* total share, not the *increase* (the "how much more" asked in the question). Always pay attention to whether the question asks for the final state or the difference between states. ### Final Answer **Therefore, the correct answer is $\frac{3D}{M^2 - 3M}$.**