The ratio of the present ages of a man and his wife is 5:8. After 10 Years, the ratio of the man and his wife will be 2:3. What will be the ratio of their ages after 20 years?

Question 5Answer

a.

09:10

b.

03:05

c.

07:10

d.

04:05

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.

The ratio of the present ages of a man and his wife is 5:8. After 10 Years, the ratio of the man and his wife will be 2:3. What will be the ratio of their ages after 20 years?

Learn how to solve a ratio-based age problem where the current ages of a man and his wife are in the ratio 5:8, and the ratio after 10 years will be 2:3. This detailed step-by-step solution walks through key concepts in algebra and proportions, perfect for students in Grades 7-9.

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