Average age of a family of three members 4 years ago is 20 years. A child was born in the family during this period. If average age of the family three years hence is 21.5 years then find the present age of the child.

Answer

To solve the problem step by step, we will follow the given information and apply the concepts of averages and ages.

### Step 1: Determine the sum of the ages of the family members 4 years ago.
Given that the average age of the family of 3 members 4 years ago was 20 years, we can find the sum of their ages at that time.

\[ \text{Average} = \frac{\text{Sum of ages}}{\text{Number of members}} \]

Let the sum of ages 4 years ago be \( S \).

\[ 20 = \frac{S}{3} \]

Multiplying both sides by 3 gives:

\[ S = 20 \times 3 = 60 \text{ years} \]

### Step 2: Calculate the present sum of ages.
Since 4 years have passed, each of the 3 members has aged 4 years. Therefore, the present sum of their ages will be:

\[ \text{Present sum} = S + (3 \times 4) = 60 + 12 = 72 \text{ years} \]

### Step 3: Determine the total age of the family 3 years hence.
We need to find the total age of the family 3 years from now. In 3 years, each of the 4 members (the 3 original members plus the newborn) will age 3 years.

\[ \text{Total age in 3 years} = 72 + (4 \times 3) = 72 + 12 = 84 \text{ years} \]

### Step 4: Set up the equation using the average age 3 years hence.
We know that the average age of the family 3 years hence is 21.5 years. Therefore, we can set up the equation:

\[ \text{Average} = \frac{\text{Total age in 3 years}}{\text{Number of members}} \]

Let \( x \) be the present age of the child. The total age in 3 years can also be expressed as:

\[ 84 = \frac{81 + x}{4} \]

### Step 5: Solve for \( x \).
Multiplying both sides by 4 gives:

\[ 84 \times 4 = 81 + x \]

Calculating the left side:

\[ 336 = 81 + x \]

Now, subtract 81 from both sides:

\[ x = 336 - 81 = 255 \]

### Step 6: Find the present age of the child.
Since we calculated the total age in 3 years, we need to find the present age of the child. The child was born during the last 4 years, so:

\[ \text{Present age of the child} = x - 3 = 255 - 3 = 2 \text{ years} \]

Thus, the present age of the child is:

\[ \text{Present age of the child} = 2 \text{ years} \]

### Summary of the Solution:
The present age of the child is **2 years**.

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