The mean age of nine boys is 28 years and if one new boy joins them the mean age increases by one.

Assertion(A): The age of new boy is (29 x 10 - 28 x 9) years.

Reason(R): The age of new boy is (29 - 28) x 10 years.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true and R is correct reason for A.

4. Both A and R are true and R is incorrect reason for A.

Given, the mean age of 9 boys is 28 years.

when one new boy joins, the mean age increases by 1 year, making the new mean 29 years

By formula,

Mean = $\dfrac{\text{Sum of all observations}}{\text{Number of all observations}}$

The mean age of 9 boys is 28 years.

$\Rightarrow \text{Mean} = \dfrac{\text{Sum of ages of 9 boys}}{\text{Number of boys}}\\[1em] \Rightarrow 28 = \dfrac{\text{Sum of ages of 9 boys}}{9}\\[1em] \Rightarrow \text{Sum of ages of the boys} = 28 \times 9\\[1em]$

After the new boy joins, the mean age becomes 29 years for 10 boys.

$\Rightarrow 29 = \dfrac{\text{Sum of ages of 10 boys}}{10}\\[1em] \Rightarrow \text{Sum of ages of 10 boys} = 29 \times 10 \\[1em]$

The age of the new boy is the difference between the total age of 10 boys and the total age of 9 boys = 29 x 10 - 28 x 9

∴ A is true, R is false.

Hence, option 1 is the correct option.