Problem 103 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem involving rate of work. A high school mathematics teacher can grade a set of chapter tests in 5 hours working alone. If her student teacher helps her, it will take them 3 hours to grade the tests. How long would it take the student teacher to grade the tests if he worked alone?

Step-by-Step Solution

Answer

The student teacher would take 7.5 hours to grade the tests alone.

Step 1: Determine Each Person's Work Rate

First, we calculate the work rate of each person working together and alone. The mathematics teacher can complete 1 set of tests in 5 hours, so her work rate is \( \frac{1}{5} \) of the tests per hour. When the student teacher helps, they finish in 3 hours, so their combined work rate is \( \frac{1}{3} \) of the tests per hour.

Step 2: Set Up the Equation

Let's denote the student teacher's work rate as \( \frac{1}{x} \) of the tests per hour. According to the given information, the combined work rate of both the teacher and the student teacher is the sum of their individual work rates. Therefore, we can write:\[ \frac{1}{5} + \frac{1}{x} = \frac{1}{3} \]

Step 3: Solve the Equation for x

To solve the equation \( \frac{1}{5} + \frac{1}{x} = \frac{1}{3} \), we first eliminate the fractions by multiplying through by 15x (the least common multiple of 5, 3, and x):\[ 15x \left( \frac{1}{5} \right) + 15x \left( \frac{1}{x} \right) = 15x \left( \frac{1}{3} \right) \]Simplifying, we get:\[ 3x + 15 = 5x \]Now solve for x:\[ 15 = 5x - 3x \]\[ 2x = 15 \]\[ x = \frac{15}{2} \]Therefore, \( x = 7.5 \).

Step 4: Interpret the Result

The result \( x = 7.5 \) indicates that if the student teacher works alone, it would take him 7.5 hours to grade the set of chapter tests.

Key Concepts

Work Rate Calculation

Understanding how quickly a person can complete a job is central to solving rate of work problems. The work rate refers to the amount of work done per unit of time.
In our exercise, we examine two individuals, a mathematics teacher and a student teacher, working at different speeds to accomplish the same task—grading tests.

• The teacher's work rate is simple to find: she can grade one set of tests alone in 5 hours, giving her a rate of \( \frac{1}{5} \) of the tests per hour.
• When working together, their combined work rate is \( \frac{1}{3} \) of the tests per hour, since they complete the task in 3 hours.

Calculating these rates is essential. It reveals how efficiently each person works and also sets the stage for solving complex group work problems.

Algebraic Equation Solving

In this problem, algebra helps us translate the word problem into a solvable equation. We introduce a variable, \( x \), to represent the student teacher's work rate, \( \frac{1}{x} \) of the tests per hour. Once we know both combined and individual work rates, we can set up an equation:

• The equation \( \frac{1}{5} + \frac{1}{x} = \frac{1}{3} \) stems from adding the individual work rates to equal the combined work rate when working together.
• To make solving easier, eliminate fractions by multiplying by the least common multiple, 15x, transforming terms into a simple linear equation: \( 3x + 15 = 5x \).

Then, solve for \( x \):

• Re-arrange to isolate \( x \) on one side, resulting in \( 2x = 15 \).
• Finally, divide through to find \( x = 7.5 \).

This method shows algebra's power in uncovering unknowns in combined work scenarios.

Time Management in Work

Effective time management allows tasks to be completed smoothly, whether individually or as a group. Solving rate of work problems teaches valuable lessons on distributing workloads to optimize performance.

• When the teacher works alone, it takes 5 hours to finish grading, demonstrating her distinct work pace.
• If only the student teacher grades, it takes 7.5 hours—a slower pace because he works alone without the teacher's assistance.

By calculating these timings, we see how splitting work between team members leads to more efficient outcomes. Understanding each person’s work rate can guide the distribution of responsibilities, enhancing productivity and decreasing completion time for shared tasks.