X,Y,W,Z jointly purchased an office space for Rs 84 lakhs. The amount contributed by Y,W and Z put together is three times of X. The amount contributed by X,W and Z put together is 320% of Y. Further, the amount contributed by W is 20% of X,Y and Z put together. What is the contribution of Z?
Step-by-step Solution:
Let the total cost be $T = X + Y + W + Z = 84$ lakhs. Given $Y + W + Z = 3X$. Adding $X$ to both sides: $$ X + Y + W + Z = 4X \implies 84 = 4X \implies X = 21 $$ Given $X + W + Z = 3.2Y$. Adding $Y$ to both sides: $$ X + Y + W + Z = 4.2Y \implies 84 = 4.2Y \implies Y = 20 $$ Given $W = 0.2(X + Y + Z) = \frac{1}{5}(X + Y + Z)$. Adding $W$ to both sides: $$ W + \frac{1}{5}W = \frac{1}{5}(X + Y + W + Z) $$ $$ \frac{6}{5}W = \frac{84}{5} \implies 6W = 84 \implies W = 14 $$ Now, substitute the values of $X$, $Y$, and $W$ to find $Z$: $$ Z = 84 - (21 + 20 + 14) $$ $$ Z = 84 - 55 = 29 \text{ lakhs} $$