Mr. Deep divides Rs. 1833 such that 4 times the 1st share, thrice the 2nd…

Concept

When several multiples of different shares are set equal (a·A = b·B = c·C), each share is inversely proportional to its multiplier. So the shares are in the ratio 1/a : 1/b : 1/c. Clearing denominators (multiply by the LCM of the multipliers) converts this into a whole-number ratio, after which the total is split into those ratio parts.

Application

1. Let the first, second and third shares be x, y and z. The condition states 4x = 3y = 2z.

2. Each share is inversely proportional to its multiplier, so x : y : z = 1/4 : 1/3 : 1/2.

3. Multiply every term by the LCM of 4, 3 and 2, which is 12: x : y : z = 12/4 : 12/3 : 12/2 = 3 : 4 : 6.

4. Total number of ratio parts = 3 + 4 + 6 = 13.

5. Value of one part = 1833 ÷ 13 = 141.

6. The second share takes 4 parts, so it equals 4 × 141 = 564.

Cross-check

With one part = 141 the three shares are 3×141 = 423, 4×141 = 564 and 6×141 = 846, and 423 + 564 + 846 = 1833 (the whole amount). The equal-multiple test also holds: 4×423 = 3×564 = 2×846 = 1692. So the second share of 564 is consistent.