Instead of dividing ₹ 117 among P, Q, R in the ratio $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$, by mistake it was divided in the ratio 2 : 3 : 4. Who gained in the transaction?

### Concept & Ratio Distribution Error To solve a distribution error problem, you must calculate the exact amounts each person received under the correct intended ratio and compare them to the amounts received under the mistaken ratio. $$ \text{Share} = \frac{\text{Ratio Part}}{\text{Total Ratio Parts}} \times \text{Total Amount} $$ ### Step-by-Step Solution 1. **Calculate the intended shares:** The intended ratio is $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$. Convert to integer ratio by multiplying by the LCM of denominators (LCM of 2, 3, 4 is 12): $(\frac{1}{2} \times 12) : (\frac{1}{3} \times 12) : (\frac{1}{4} \times 12) = 6 : 4 : 3$. Total intended parts = $6 + 4 + 3 = 13$. Total sum = ₹ 117. Multiplier = $117 \div 13 = 9$. P's intended share = $6 \times 9 = 54$ Q's intended share = $4 \times 9 = 36$ R's intended share = $3 \times 9 = 27$ 2. **Calculate the mistaken shares:** The mistaken ratio is $2 : 3 : 4$. Total mistaken parts = $2 + 3 + 4 = 9$. Multiplier = $117 \div 9 = 13$. P's mistaken share = $2 \times 13 = 26$ Q's mistaken share = $3 \times 13 = 39$ R's mistaken share = $4 \times 13 = 52$ 3. **Compare intended vs. actual:** P went from 54 to 26 (Loss) Q went from 36 to 39 (Gain) R went from 27 to 52 (Gain) ### Exam Strategy & Shortcut Look at the ratios comparatively. The original ratio ($6:4:3$) heavily favors P. The new ratio ($2:3:4$) heavily favors R, and slightly favors Q compared to the original relative distribution. You can often deduce that the people whose relative weighting increased (Q and R) will be the gainers without calculating exact rupee values, saving time. ### Common Pitfall A common pitfall is forgetting to convert the initial fractional ratio into an integer ratio, leading to incorrect base values for comparison. ### Final Answer Therefore, the correct answer is **Both Q and R**.