The ratio of the ages of a man and his wife is 4 : 3. After 4 years, this ratio will be 9 : 7. If at the time of their marriage, the ratio of their ages was 5 : 3, then how many years ago were they married?

### Concept & Strategy This is a three-stage timeline problem (Marriage $\rightarrow$ Present $\rightarrow$ Future). First, solve for the present ages using the present-to-future transition, then use those established ages to find the historical marriage timeline. $$ \frac{\text{Present}_A + t}{\text{Present}_B + t} = \text{Future Ratio} $$ ### Step-by-Step Solution * **Given:** Present ratio = $4 : 3$. Ratio in 4 years = $9 : 7$. Marriage ratio = $5 : 3$. * **Calculation:** Let the present ages of the man and wife be $4x$ and $3x$. * Establish the future equation: $\frac{4x + 4}{3x + 4} = \frac{9}{7}$ * Cross-multiply to solve for $x$: $7(4x + 4) = 9(3x + 4)$ * Expand: $28x + 28 = 27x + 36$ * Isolate $x$: $x = 8$. * Calculate their exact present ages: Man = $4(8) = 32$ years. Wife = $3(8) = 24$ years. * Now, let $y$ be the number of years ago they got married. * Set up the past ratio equation: $\frac{32 - y}{24 - y} = \frac{5}{3}$ * Cross-multiply again: $3(32 - y) = 5(24 - y)$ * Expand: $96 - 3y = 120 - 5y$ * Solve for $y$: $2y = 24 \implies y = 12$. ### Exam Strategy & Shortcut **Cross-Product Method:** For the present-to-future step: Present = $4 : 3$ Future ($+4$) = $9 : 7$ Difference in cross products = $(9 \times 3) - (7 \times 4) = 27 - 28 = 1$ unit. Difference in time cross products = $(9 \times 4) - (7 \times 4) = 36 - 28 = 8$ years. So, 1 unit = 8 years. Present ages are $32$ and $24$. Check options for marriage $y$ years ago. If $y=12$, ages were $20$ and $12$, which simplifies exactly to $5 : 3$. ### Common Pitfall Getting bogged down trying to create a massive single equation with both $x$ and $y$ simultaneously. Break the problem into two distinct, manageable phases: find present ages first, then find the marriage date. ### Final Answer Therefore, the correct answer is **12 years**.