A sum of ₹ 1,710 is divided in A, B and C such that 4 times of A, 6 times of B…

Concept

When several multiples of unknown shares are set equal (here 4 times A = 6 times B = 9 times C), set that common value to a constant k. Each share then equals k divided by its multiplier, so the shares are inversely proportional to their multipliers. Converting these to whole-number ratio parts lets the total be split proportionally.

Application

1. Let 4A = 6B = 9C = k. Then A = k/4, B = k/6, C = k/9, so A : B : C = 1/4 : 1/6 : 1/9.

2. Multiply each term by the LCM of 4, 6, 9 (which is 36): A : B : C = 9 : 6 : 4.

3. Total ratio parts = 9 + 6 + 4 = 19. One part = 1710 ÷ 19 = 90.

4. A = 9 × 90 = 810, B = 6 × 90 = 540, C = 4 × 90 = 360.

5. Difference between A and C = 810 − 360 = 450.

Cross-check

Add the shares back: 810 + 540 + 360 = 1710, matching the total. Also 4 × 810 = 3240, 6 × 540 = 3240, 9 × 360 = 3240 — all equal, confirming the split. Hence A − C = ₹ 450.