Let the present ages of the man and his wife be $5x$ and $8x$ respectively, based on the ratio given as 5:8.

Step 1: Establish equations based on future age conditions

In 10 years, the man’s age will be $5x + 10$ and the wife’s age will be $8x + 10$.

According to the problem, after 10 years, the ratio of their ages will be 2:3. So, we can set up the equation:

$\frac{5x + 10}{8x + 10} = \frac{2}{3}$

Step 2: Solve the equation

Cross-multiply to eliminate the fractions:

$3(5x + 10) = 2(8x + 10)$

Simplify both sides:

$15x + 30 = 16x + 20$

Now, solve for $x$:

$15x - 16x = 20 - 30$ $-x = -10$ $x = 10$

Step 3: Find the present ages

Now that we know $x = 10$, we can calculate the present ages:

• Man's present age: $5x = 5(10) = 50$
• Wife's present age: $8x = 8(10) = 80$

Step 4: Calculate their ages after 20 years

After 20 years, the man’s age will be: $50 + 20 = 70$ And the wife’s age will be: $80 + 20 = 100$

Step 5: Find the ratio of their ages after 20 years

The ratio of their ages after 20 years is: $\frac{70}{100} = \frac{7}{10}$

Thus, the ratio of their ages after 20 years will be 7:10.

Final Answer:

The correct option is c. 07:10

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Would you like more details or have any questions? Here are five related questions to consider:

1. How can you determine ratios in real-life scenarios like income or property distribution?
2. How do ratios change over time, and how do they apply in age-related problems?
3. What are some common mistakes made when solving ratio-based age problems?
4. How can cross-multiplication help in solving proportion equations?
5. How do ratios compare to percentages in terms of representing relationships?

Tip: Always set up clear equations from the word problem before solving.