Question

If a woman sleeps only 6 hours per night and spends \(\frac{4}{9}\) of her waking hours at work, then what fraction of the total 24-hour day (on a work day) is the woman at work? A. \(\frac{1}{9}\) B. \(\frac{1}{3}\) C. \(\frac{4}{9}\) D. \(\frac{3}{4}\)

Short answer

The woman spends \(\frac{1}{3}\) of the total 24-hour day at work. The correct answer is B. \(\frac{1}{3}\).

Step-by-step solution

Calculate the total waking hours

The woman sleeps 6 hours per night, so she is awake for the remaining 18 hours of the 24-hour day.

Calculate the hours spent at work

Next, we need to determine how many of these waking hours are spent at work. Since she spends \(\frac{4}{9}\) of her waking hours at work, we can calculate the number of hours spent at work by multiplying her waking hours (18) by \(\frac{4}{9}\): \(Hours \, at \, work = 18 \cdot \frac{4}{9} \)

Simplify the calculation

To simplify the multiplication, we can cancel the factor of 9 in both the numerator and denominator: \(Hours \, at \, work= 2 \cdot \frac{4}{1} \) Now, multiply to find the number of hours she spends at work: \(Hours \, at \, work = 2 \cdot 4 = 8\) The woman spends 8 hours at work.

Determine the fraction of the 24-hour day she is at work

To determine the fraction of the 24-hour day the woman is at work, we can simply divide her hours spent at work (8 hours) by the total number of hours in a day (24 hours): \(Fraction \, of \, day \, at \, work = \frac{8}{24} \) Now, simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 8: \(Fraction \, of \, day \, at \, work = \frac{1}{3} \) So, the woman spends \(\frac{1}{3}\) of the total 24-hour day at work. The correct answer is B. \(\frac{1}{3}\).