In a certain company, the ratio of the number of managers to the number of production-line workers is 5 to 72. If 8 additional production-line workers were to be hired, the ratio of the number of managers to the production-line workers would be 5 to 74. How many managers does the company have? (A.T.M.A., 2006)

### Concept & Direct Ratio Comparison When the quantity of only one component in a ratio changes, you can often compare the "parts" directly if the unchanged component's ratio value remains identical in both the old and new ratios. ### Step-by-Step Solution * **Step 1: Analyze the ratios.** * Initial ratio of Managers to Workers = $5 : 72$ * Final ratio of Managers to Workers = $5 : 74$ * Notice that the ratio parts for Managers remained $5$ in both scenarios. * **Step 2: Compare the changing parts.** * The parts for Workers increased from $72$ to $74$. * Increase in parts = $74 - 72 = 2$ parts. * **Step 3: Equate parts to real values.** * This $2$-part increase corresponds exactly to the $8$ additional workers hired. $$ 2 \text{ parts} = 8 \text{ workers} $$ $$ 1 \text{ part} = 4 \text{ workers} $$ * **Step 4: Find the number of managers.** * Managers represent $5$ parts. * Number of managers = $5 \times 4 = 20$. ### Exam Strategy & Shortcut This is a classic "no-pen" question. You see the managers' ratio part ($5$) is constant. The workers' ratio part jumps by $2$. Since $2$ parts $= 8$ real workers, the multiplier is $4$. Therefore, managers $= 5 \times 4 = 20$. Solved in 5 seconds mentally. ### Common Pitfall Setting up full cross-multiplication equations $\frac{5x}{72x + 8} = \frac{5}{74}$ is mathematically correct but wastes precious time since the numerator $5$ cancels out immediately. ### Final Answer Therefore, the correct answer is **20**.