When a sum of money was distributed to 12 persons instead of 16 persons, each person got ₹ 400 more. What was the sum?

## Concept & Logic This problem involves inverse variation. The total sum is constant. The difference in the per-person share between the two scenarios forms a simple algebraic equation based on fractions of the total sum. $$ \frac{Total}{Smaller\ Group} - \frac{Total}{Larger\ Group} = Difference\ in\ Share $$ ## Step-by-Step Solution * **Given:** Let the total sum of money be $x$. Original scenario: Divided among 16 persons. Share = $\frac{x}{16}$. New scenario: Divided among 12 persons. Share = $\frac{x}{12}$. Difference in share = ₹ 400. * **Calculation:** Set up the equation representing the difference: $\frac{x}{12} - \frac{x}{16} = 400$ * **Calculation:** Find the Lowest Common Multiple (LCM) of 12 and 16, which is 48. Rewrite the fractions: $\frac{4x}{48} - \frac{3x}{48} = 400$ $\frac{x}{48} = 400$ * **Calculation:** Solve for $x$: $x = 400 \times 48 = 19200$. ## Exam Strategy & Shortcut Use ratio mapping to avoid algebra. The ratio of people is $12 : 16$, which simplifies to $3 : 4$. Because the sum is constant, the ratio of their individual shares is inversely proportional: $4 : 3$. The difference in shares is 1 unit ($4 - 3 = 1$ unit). We are told this difference is ₹ 400. So, 1 unit = ₹ 400. The share when divided among 12 people (the 4-unit share) is $4 \times 400 = 1600$. Total sum = Share $\times$ Number of people = $1600 \times 12 = 19200$. ## Common Pitfall Writing the equation backwards as $\frac{x}{16} - \frac{x}{12} = 400$, which results in a negative value for the total sum. Always remember that dividing by a smaller number yields a larger share, so $\frac{x}{12}$ must come first. ## Final Answer Therefore, the correct answer is **₹ 19200**.