Question

A total of $22$ men and $26$ women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly $35$ years. If the average age of the men was exactly $38$ years, which of the following was closest to the average age, in years of the women?

• A. $31$
• B. $31.5$
• C. $32$
• D. $32.5$
• E. $33$

Answer 1

The correct option is D $32.5$

Mean is the sum of the values in the data set divided by the total number of values in the data set.

Given that mean of $22$ men and $26$ women ages is $35$ years.

Let the total age of $22$ men is $M_1$ and

the total age of $26$ women is $W_1$

Therefore, $\frac{M_1+W_1}{22+26}=35$

$⟹\frac{M_1+W_1}{48}=35$
$⟹M_1+W_1=35\times 48$

$⟹M_1+W_1=1680$ -----(1)

Given that the average age of men is $38$ years.

$⟹\frac{M_1}{22}=38$

$⟹M_1=38\times 22$
$⟹M_1=836$ ----(2)

On substituting (2) in (1) we get

$836+W_1=1680$

$⟹W_1=1680-836=844$

Therefore, average age of the women is $\frac{W_1}{26}=\frac{844}{26}=32.5$
Hence, average age of women is $32.5$.