In 1998, the respective ratio of ages of father, mother and their only son was 6:5:2. In 2002, a girl (daughter) was born. If in 2003, the average age of 4 members of the family was 30 years, what will be the son's age in 2021?

Options

A. 39
B. 41
C. 43
D. 45

Answer

To solve the problem step by step, we will follow these calculations:

1. **Understanding the Ratios**:
In 1998, the ages of the father, mother, and son are in the ratio of 6:5:2. Let's denote their ages as:
- Father's age = 6x
- Mother's age = 5x
- Son's age = 2x

2. **Calculating the Total Age in 1998**:
The total age of the family in 1998 can be expressed as:
\[ \text{Total age} = 6x + 5x + 2x = 13x \]

3. **Ages in 2003**:
By 2003, 5 years have passed since 1998. Therefore, the ages of each family member would be:
- Father's age in 2003 = 6x + 5
- Mother's age in 2003 = 5x + 5
- Son's age in 2003 = 2x + 5
- Daughter's age in 2003 = 0 (since she was born in 2002)

4. **Average Age Calculation**:
We know the average age of the four family members in 2003 is 30 years. Thus, we can set up the equation:
\[ \frac{(6x + 5) + (5x + 5) + (2x + 5) + 0}{4} = 30 \]
Simplifying the left side:
\[ \frac{(6x + 5 + 5x + 5 + 2x + 5)}{4} = 30 \]
\[ \frac{(13x + 15)}{4} = 30 \]

5. **Solving for x**:
Multiply both sides by 4:
\[ 13x + 15 = 120 \]
Subtract 15 from both sides:
\[ 13x = 105 \]
Divide by 13:
\[ x = \frac{105}{13} = 8.0769 \approx 8 \]

6. **Finding Son's Age in 1998**:
The son's age in 1998 is:
\[ \text{Son's age} = 2x = 2 \times 8 = 16 \text{ years} \]

7. **Calculating Son's Age in 2021**:
From 1998 to 2021 is 23 years. Therefore, the son's age in 2021 will be:
\[ \text{Son's age in 2021} = 16 + 23 = 39 \text{ years} \]

Thus, the son's age in 2021 is **39 years**.