If $\frac{a}{x} + \frac{y}{b} = 1$ and $\frac{b}{y} + \frac{z}{c} = 1$, then $\frac{x}{a} + \frac{c}{z}$ will be equal to:

### Concept & Strategy This is a cyclic equation problem. The key is to isolate the inverse of the required terms ($\frac{x}{a}$ and $\frac{c}{z}$) from the given equations and express them in terms of the common linking variables ($y$ and $b$). ### Step-by-Step Solution Given Equations: 1) $\frac{a}{x} + \frac{y}{b} = 1$ 2) $\frac{b}{y} + \frac{z}{c} = 1$ We need to find the value of: $\frac{x}{a} + \frac{c}{z}$ From equation (1), isolate $\frac{a}{x}$: $\frac{a}{x} = 1 - \frac{y}{b}$ $\frac{a}{x} = \frac{b - y}{b}$ Taking the reciprocal to find $\frac{x}{a}$: $\frac{x}{a} = \frac{b}{b - y}$ From equation (2), isolate $\frac{z}{c}$: $\frac{z}{c} = 1 - \frac{b}{y}$ $\frac{z}{c} = \frac{y - b}{y}$ Taking the reciprocal to find $\frac{c}{z}$: $\frac{c}{z} = \frac{y}{y - b}$ Now, substitute these into the expression we need to evaluate: $\frac{x}{a} + \frac{c}{z} = \frac{b}{b - y} + \frac{y}{y - b}$ To make the denominators the same, rewrite $\frac{y}{y - b}$ as $-\frac{y}{b - y}$: $= \frac{b}{b - y} - \frac{y}{b - y}$ $= \frac{b - y}{b - y}$ $= 1$ ### Exam Strategy & Shortcut **Value Putting Method:** Assign values to variables that satisfy the given equations. Let $\frac{y}{b} = 2$. This means $\frac{b}{y} = 0.5$. From Eq 1: $\frac{a}{x} + 2 = 1 \implies \frac{a}{x} = -1 \implies \frac{x}{a} = -1$. From Eq 2: $0.5 + \frac{z}{c} = 1 \implies \frac{z}{c} = 0.5 \implies \frac{c}{z} = 2$. Now find $\frac{x}{a} + \frac{c}{z}$: $-1 + 2 = 1$. ### Common Pitfall A major trap is attempting to solve for individual variables ($a, b, c, x, y, z$). Because there are 6 variables and only 2 equations, finding individual values is impossible. Always treat grouped fractions as single units. ### Final Answer Therefore, the correct answer is **1**.