Let the number be $x$.
First term: $\frac{4}{15} \times \frac{5}{7} x = \frac{4}{21} x$.
Second term: $\frac{4}{9} \times \frac{2}{5} x = \frac{8}{45} x$.

Given difference:
$\frac{4}{21} x - \frac{8}{45} x = 8$
LCM of 21 and 45 = 315.
$\frac{15 \times 4 - 7 \times 8}{315} x = 8$
$\frac{60 - 56}{315} x = 8 \implies \frac{4}{315} x = 8 \implies x = \frac{8 \times 315}{4} = 2 \times 315 = 630$.

The question asks for half of that number:
$\frac{x}{2} = \frac{630}{2} = 315$.

Option analysis:

• 112: Calculation error.
• 216: Calculation error.
• 315: Correct half of the number.
• 412: Calculation error.

The average of the first $n$ natural numbers ($1, 2, 3, \dots, n$) is given by:
$\text{Average} = \frac{n + 1}{2}$
For $n = 100$:
$\text{Average} = \frac{100 + 1}{2} = \frac{101}{2} = 50.5$.

Option analysis:

• 50: Ignores the $+1$ term in median calculation.
• 50.5: Correct average of numbers 1 to 100.
• 51: Calculation error.
• 49.5: Average of numbers 0 to 99.