The average age of 4 members of a family is 20 yr . If youngest member is 4 yr old, what was the average age of family at the time of the birth of youngest member?

Options

A. 27 yr
B. `23(1)/(3) yr `
C. `21 (1)/(3)`
D. 22 yr

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step-by-Step Solution:

1. **Understand the Problem**: We know that the average age of 4 family members is 20 years, and the youngest member is currently 4 years old. We need to find the average age of the family at the time the youngest member was born.

2. **Calculate the Total Age of the Family**:
- The average age of the family is given by the formula:
\[ \text{Average} = \frac{\text{Sum of ages}}{\text{Number of members}} \]
- Here, the average age is 20 years and there are 4 members. Therefore, we can set up the equation:
\[ 20 = \frac{f_1 + f_2 + f_3 + f_4}{4} \]
- Multiplying both sides by 4 gives:
\[ f_1 + f_2 + f_3 + f_4 = 80 \]

3. **Subtract the Youngest Member's Age**:
- Since the youngest member is 4 years old, we can find the sum of the ages of the other three members:
\[ f_1 + f_2 + f_3 = 80 - 4 = 76 \]

4. **Determine Ages at the Time of the Youngest Member's Birth**:
- The youngest member is currently 4 years old, meaning they were born 4 years ago. At that time, all other family members were 4 years younger:
\[ f_1 - 4, f_2 - 4, f_3 - 4 \]

5. **Calculate the Total Age of the Family at that Time**:
- The total age of the family members at the time of the youngest member's birth would be:
\[ (f_1 - 4) + (f_2 - 4) + (f_3 - 4) = (f_1 + f_2 + f_3) - 12 \]
- Substituting the sum we found earlier:
\[ = 76 - 12 = 64 \]

6. **Calculate the Average Age at that Time**:
- The average age of the three members at that time would be:
\[ \text{Average} = \frac{64}{3} \]
- Performing the division:
\[ 64 \div 3 = 21.33 \text{ or } 21 \frac{1}{3} \text{ years} \]

### Final Answer:
The average age of the family at the time of the birth of the youngest member was \( 21 \frac{1}{3} \) years.

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