Rs 1980 is distributed among X, Y and Z such that X gets 5/6 of what Y and Z…

Concept:

When a share (say A) is given as a fraction k/m of the SUM of the other two shares (B+C), and the TOTAL of all three is known, the fastest route is to replace (B+C) with (Total - A) inside that fraction. This turns a three-variable condition into a single linear equation in A alone, which can be solved directly without first finding the other two shares.

Application:

1. Since Y+Z = 1980 - X, substitute into X = 5/6(Y+Z): X = 5/6(1980 - X). This gives 6X = 5(1980-X), so 6X = 9900 - 5X, so 11X = 9900, giving X = Rs 900.

2. Since X+Y = 1980 - Z, substitute into Z = 7/15(X+Y): Z = 7/15(1980 - Z). This gives 15Z = 7(1980-Z), so 15Z = 13860 - 7Z, so 22Z = 13860, giving Z = Rs 630.

3. Y is what remains of the total once X and Z are removed: Y = 1980 - X - Z = 1980 - 900 - 630 = Rs 450.

Cross-check:

Substituting back: X = 5/6 of (Y+Z) = 5/6 of (450+630) = 5/6 of 1080 = 900, which matches. Z = 7/15 of (X+Y) = 7/15 of (900+450) = 7/15 of 1350 = 630, which matches. And 900 + 450 + 630 = 1980, the given total. Both conditions and the total are satisfied, confirming Y's share is Rs 450.