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. \(\frac{1}{3}\).

Key concepts

Fractions

Fractions are a fundamental mathematical concept that allows us to represent a part of a whole. In this exercise, the fraction \(\frac{4}{9}\) is used to express the portion of the woman's waking hours that she spends at work. A fraction consists of a numerator and a denominator. The numerator (top number) represents how many parts we have, and the denominator (bottom number) represents how many equal parts the whole is divided into.

When calculating how much of her waking time the woman spends at work, multiplying the total waking hours (18) by \(\frac{4}{9}\) gives us the actual number of hours spent at work. The calculation involves:

• Multiplying the whole number by the fraction: \(18 \times \frac{4}{9}\).
• Simplifying by dividing common factors in the numerator and denominator.

Understanding fractions is crucial because it enables you to convert partial values into whole numbers easily and vice versa, helping in various real-life scenarios like sharing, budgeting, and numerous mathematical computations.

Working Hours

Working hours refer to the amount of time someone dedicates to their job during the day. In this problem, the woman's working hours are calculated as a fraction of her waking hours, demonstrating the relationship between her work-life and personal life. The exercise simplifies this by considering only the time when she is not asleep, which is 18 hours out of the total 24-hour day.

By determining that she spends \(\frac{4}{9}\) of these waking hours at work, the exercise emphasizes the importance of calculating proportions in our daily schedules. Given her waking hours are 18, the calculation \(18 \times \frac{4}{9}\) results in 8 hours worked. This real-world application shows how we can use mathematical concepts, like fractions, to acquire clear insights into our daily time allocation requirements and engage in effective scheduling.

Time Management

Time management is a vital skill, especially when balancing work, rest, and personal activities. The exercise showcases it by examining the woman's time allocation throughout her day. She spends 6 hours sleeping, which leaves 18 hours for other activities, including work and personal time.

Good time management involves effectively allocating your available hours to enhance productivity while ensuring sufficient rest. Understanding the fraction \(\frac{1}{3}\) of her total day spent at work can help her maintain a healthy work-life balance by planning her non-working hours wisely. Time management strategies could include:

• Prioritizing tasks during non-working hours.
• Ensuring enough time for rest and leisure to prevent burnout.
• Understanding your daily time commitments through accurate calculations like the ones displayed in this exercise.

By using simple mathematical tools, you can gain more control over your daily routine and life efficiency, which is essential for success in both professional and personal spheres.