Q 8 - If 100 year is equal to the sum of the ages of father and son. 2:1 was the ratio of father and son before the period of 5 years. Find out the ratio of ages which would be after the period 10 year.

A - 3:4

To solve this problem, we need to determine the ages of the father and son based on the given information, then find their ages after 10 years and compute the ratio of their ages at that time.

### Step 1: Set Up Equations Based on the Information

1. **Given Information:**
   - The sum of the ages of the father and son is 100 years.
   - The ratio of their ages was 2:1 five years ago.

Let's denote:
- The father's current age as \( F \)
- The son's current age as \( S \)

2. **Equation from the sum of ages:**
   \[
   F + S = 100
   \]

3. **Equation from the ratio 5 years ago:**
   Five years ago, the father's age was \( F - 5 \) and the son's age was \( S - 5 \). The ratio of their ages was 2:1. So,
   \[
   \frac{F - 5}{S - 5} = 2
   \]

### Step 2: Solve the Equations

4. From the ratio equation:
   \[
   F - 5 = 2(S - 5)
   \]
   Expand and simplify:
   \[
   F - 5 = 2S - 10
   \]
   \[
   F = 2S - 5
   \]

5. Substitute this expression for \( F \) into the sum of ages equation:
   \[
   (2S - 5) + S = 100
   \]
   \[
   3S - 5 = 100
   \]
   \[
   3S = 105
   \]
   \[
   S = 35
   \]

6. Now, find \( F \):
   \[
   F = 2S - 5 = 2(35) - 5 = 70 - 5 = 65
   \]

So, the current ages are:
- Father's age \( F = 65 \) years
- Son's age \( S = 35 \) years

### Step 3: Determine Ages After 10 Years

7. After 10 years:
   - Father's age will be \( 65 + 10 = 75 \) years.
   - Son's age will be \( 35 + 10 = 45 \) years.

### Step 4: Calculate the Ratio of Their Ages After 10 Years

8. The ratio of the father's age to the son's age after 10 years is:
   \[
   \text{Ratio} = \frac{75}{45}
   \]

9. Simplify the ratio by dividing both numerator and denominator by 15:
   \[
   \frac{75 \div 15}{45 \div 15} = \frac{5}{3}
   \]

Therefore, the ratio of the ages of the father and son after 10 years will be **5:3**, not **3:4**.

### Conclusion:
There seems to be an error in the provided answer "3:4." The correct ratio of their ages after 10 years is **5:3**.

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Would you like further details on any part of the solution, or do you have any questions?

### Related Questions:
1. How would the solution change if the initial age sum was different?
2. What would the ratio be if the ages were 3:2 five years ago?
3. Can we generalize a formula for such age problems?
4. How do we solve if more variables (like more family members) are introduced?
5. What are other methods to simplify ratios effectively?

**Tip:** When dealing with ratios and age problems, always translate word problems into equations and solve step-by-step.

Q 8 - If 100 year is equal to the sum of the ages of father and son. 2:1 was the ratio of father and son before the period of 5 years. Find out the ratio of ages which would be after the period 10 years. A - 3:4

Learn how to solve an age problem where the sum of a father and son's ages is 100 years, and the ratio of their ages was 2:1 five years ago. This detailed step-by-step solution includes algebraic equations and ratio simplification, suitable for students in Grades 8-10.

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