Question

If 1/3 of an office staff is on vacation and 1/4 of those not on vacation are out sick, what fraction of the staff is in the office

Answer 1

Let x be the total number of staff in the office.

Thus,

The number of staff on vacation = $\frac{1}{3} x$

Since, it says that staff out sick are among those not on vacation, we have to subtract the number of staff on vacation from the total number of staff.

The number of staff out sick = $\frac{1}{4} (x - \frac{1}{3}x)$

Hence,

The number of staff in the office = $x - \frac{1}{3} x - \frac{1}{4} (x- \frac{1}{3} x) = (x - \frac{1}{3} x)(1 - \frac{1}{4} ) = x(1- \frac{1}{3})(1- \frac{1}{4}) = x( \frac{2}{3})( \frac{3}{4})$ $= \frac{1}{4} x$