The average age of a group of 6 boys is 19. If the new average age after a boy joins the group is 21.28, find the age of new boy.

Options

A. Approx 23
B. Approx 32
C. Approx 35
D. none of these

Answer

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

### Step 1: Calculate the total age of the 6 boys.
The average age of the 6 boys is given as 19 years. To find the total age of these boys, we can use the formula for average:

\[ \text{Total Age} = \text{Average Age} \times \text{Number of Boys} \]

So, we calculate:

\[ \text{Total Age of 6 boys} = 19 \times 6 = 114 \text{ years} \]

### Step 2: Set up the equation for the new average age after the new boy joins.
When a new boy joins the group, the total number of boys becomes 7. The new average age is given as 21.28 years. We can set up the equation for the total age of the 7 boys:

\[ \text{New Total Age} = \text{Total Age of 6 boys} + \text{Age of new boy} \]

Let the age of the new boy be \( X \). Thus, we can write:

\[ \text{New Total Age} = 114 + X \]

### Step 3: Write the equation for the new average.
The average age of the 7 boys can also be expressed as:

\[ \text{New Average} = \frac{\text{New Total Age}}{\text{Number of Boys}} = \frac{114 + X}{7} \]

Setting this equal to the new average age:

\[ \frac{114 + X}{7} = 21.28 \]

### Step 4: Solve for \( X \).
To eliminate the fraction, multiply both sides by 7:

\[ 114 + X = 21.28 \times 7 \]

Calculating the right side:

\[ 21.28 \times 7 = 148.96 \]

Now we have:

\[ 114 + X = 148.96 \]

Subtracting 114 from both sides gives:

\[ X = 148.96 - 114 \]

Calculating that:

\[ X = 34.96 \]

### Step 5: Round the age of the new boy.
Since age is typically expressed in whole numbers, we can round 34.96 to the nearest whole number:

\[ X \approx 35 \]

Thus, the age of the new boy is approximately **35 years**.

### Final Answer:
The age of the new boy is **35 years**.

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