If $x = 5$, $y = 3$ and $z = 2$, then $\frac{x(y - z)}{y(x + y + z)} = $

### Concept & Formula This is a direct algebraic substitution problem. The strategy is to substitute the given numerical values into the rational expression and simplify using basic arithmetic. Expression to evaluate: $$\frac{x(y - z)}{y(x + y + z)}$$ ### Step-by-Step Solution * **Step 1: Substitute the given values** Given: $x = 5$, $y = 3$, $z = 2$. Replace the variables in the expression: $$\frac{5(3 - 2)}{3(5 + 3 + 2)}$$ * **Step 2: Simplify the Parentheses** Numerator parenthesis: $(3 - 2) = 1$ Denominator parenthesis: $(5 + 3 + 2) = 10$ * **Step 3: Multiply the terms** Now substitute the resolved parentheses back into the fraction: $$\text{Numerator} = 5 \times 1 = 5$$ $$\text{Denominator} = 3 \times 10 = 30$$ $$\text{Fraction} = \frac{5}{30}$$ * **Step 4: Reduce the fraction** Both numbers are divisible by $5$: $$\frac{5 \div 5}{30 \div 5} = \frac{1}{6}$$ ### Exam Strategy & Shortcut Look for early cancellations before multiplying the denominator. Once you resolve the parentheses, you have $\frac{5 \times 1}{3 \times 10}$. You can immediately recognize that $5$ goes into $10$ exactly $2$ times. Cross out the $5$ and the $10$, leaving a $2$ in the denominator: $\frac{1}{3 \times 2} = \frac{1}{6}$. This minimizes arithmetic and prevents calculation errors on larger numbers. ### Common Pitfall A common error is erroneously "canceling out" terms inside the parentheses without evaluating them completely (e.g., trying to cancel the $y$ in the numerator's bracket with the $y$ in the denominator). Always fully resolve addition and subtraction within brackets before attempting to cancel or reduce fractions. ### Final Answer **Therefore, the correct answer is 1/6.**