Divide ₹ 671 among $A, B$ and $C$ such that if their shares be increased by ₹ 3, ₹ 7 and ₹ 9 respectively the remainder shall be in the ratio $1 : 2 : 3$.

### Concept & Adjusted Total When modifications (increases or decreases) are applied to individual shares to reach a specific ratio, apply those same modifications to the total sum first to find the new distribution base. ### Step-by-Step Solution Given: Total initial amount = ₹ 671 Increases: $A$ by ₹ 3, $B$ by ₹ 7, $C$ by ₹ 9 New ratio of shares = $1 : 2 : 3$ 1. Calculate the new total sum after the virtual increases are added: New Total = Original Total + Sum of increases New Total = $671 + (3 + 7 + 9) = 671 + 19 = 690$ 2. Distribute this new total (₹ 690) according to the new ratio ($1 : 2 : 3$). Total ratio parts = $1 + 2 + 3 = 6$ Value of 1 part = $\frac{690}{6} = 115$ 3. Calculate the increased shares for each: Increased share of $A = 1 \text{ part} = 115$ Increased share of $B = 2 \text{ parts} = 230$ Increased share of $C = 3 \text{ parts} = 345$ 4. Find the original shares by subtracting the respective increases: Original share of $A = 115 - 3 = 112$ Original share of $B = 230 - 7 = 223$ Original share of $C = 345 - 9 = 336$ ### Exam Strategy & Shortcut You can test the options directly. Take option (b): $112, 223, 336$. Add the increases: $112+3=115$, $223+7=230$, $336+9=345$. Check the ratio: $115 : 230 : 345$. This clearly simplifies to $1 : 2 : 3$. Also, verify the sum $112+223+336 = 671$. Both conditions met. ### Common Pitfall Distributing the original total (₹ 671) in the ratio of $1:2:3$ and then subtracting the amounts, failing to realize the ratio applies only *after* the total pool has been increased. ### Final Answer Therefore, the correct answer is **₹ 112, ₹ 223, ₹ 336**.