Setup the Mean equation

The mean formula for grouped data is $\overline{x} = \frac{\sum f_i x_i}{\sum f_i}$. Given $\overline{x} = 40$, we express the numerator and denominator.

$40 = \frac{(10 \times 21) + (25 \times X) + (70 \times 22) + (85 \times 35)}{21 + X + 22 + 35}$

Simplify the Numerator and Denominator

Calculating the products: $10 \times 21 = 210$, $70 \times 22 = 1540$, and $85 \times 35 = 2975$. Summing these: $210 + 1540 + 2975 = 4725$. The denominator is $21 + 22 + 35 + X = 78 + X$.

$40 = \frac{4725 + 25X}{78 + X}$

Solve for X

Multiply both sides by $(78 + X)$ to clear the fraction: $40(78 + X) = 4725 + 25X$. Expanding gives $3120 + 40X = 4725 + 25X$.

$40X - 25X = 4725 - 3120$

Final Calculation

Subtracting terms gives $15X = 1605$. Dividing by 15 yields $X = 107$.

$X = \frac{1605}{15} = 107$