Reconstruct original amounts from shifted ratio: ₹ 2186 is distributed among A, B, and C. If their amounts, after subtracting ₹ 26, ₹ 28, and ₹ 32 respectively, are in the ratio 9 : 13 : 8, find the amount originally given to A.

Introduction / Context:When a ratio applies after fixed adjustments, translate the ratio into variables for the post-change amounts, then add back the adjustments to obtain originals. Use the total to determine the scale factor.

Given Data / Assumptions:

• Total = ₹ 2186.
• After changes: (A − 26) : (B − 28) : (C − 32) = 9 : 13 : 8.

Concept / Approach:Let A − 26 = 9t, B − 28 = 13t, C − 32 = 8t. Then A = 9t + 26, B = 13t + 28, C = 8t + 32. Sum equals the total and yields t. Substitute back to find A.

Step-by-Step Solution:A + B + C = (9t + 26) + (13t + 28) + (8t + 32) = 30t + 86.30t + 86 = 2186 ⇒ 30t = 2100 ⇒ t = 70.A = 9t + 26 = 630 + 26 = ₹ 656.

Verification / Alternative check:Check post-change: A − 26 = 630, B − 28 = 910, C − 32 = 560 ⇒ ratio 630 : 910 : 560 = 9 : 13 : 8 (divide by 70). Sum of originals: 656 + 938 + 592 = 2186 confirms the total.

Why Other Options Are Wrong:

• ₹ 575, ₹ 640, and ₹ 672 do not satisfy the total with the given post-change ratio.

Common Pitfalls:

• Mistaking the 9 : 13 : 8 as the original ratio rather than the ratio after subtraction.

Final Answer:₹ 656