Problem 26 - Beginning and Intermediate Algebra

Question

It takes \(m\) hours to grade a set of papers. (a) What is the grader's rate (in job per hour)? (b) How much of the job will the grader do in \(2 \mathrm{hr} ?\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{m}\) job per hour. (b) \(\frac{2}{m} \) of the job.

Step 1: Identify Given Information

It is given that it takes the grader \(m\) hours to grade a set of papers. This is the total time taken to complete the job.

Step 2: Determine Grader's Rate

The grader's rate (in job per hour) is calculated as the reciprocal of the time taken to complete the job. Therefore, the rate is \[ \text{Rate} = \frac{1}{m} \text{ job per hour} \]

Step 3: Calculate Job Completed in 2 Hours

To find out how much of the job the grader will complete in 2 hours, multiply the rate by the time (2 hours). \[ \text{Job completed} = \text{Rate} \times 2 = \frac{1}{m} \times 2 = \frac{2}{m} \]

Key Concepts

Understanding Reciprocals

A reciprocal of a number is simply flipping that number. If you have a fraction, the reciprocal is created by swapping its numerator and denominator. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
For whole numbers, you can think of them as having an implicit denominator of 1. So, the reciprocal of 5 is \( \frac{1}{5} \).
Reciprocals are especially important in rate problems like this exercise.
When we talk about a rate, what we’re essentially looking at is how much work or task is done over a certain period. Thus, for a task that takes \( m \) hours, the rate becomes \( \frac{1}{m} \) jobs per hour.

Time Rate Work Problems

Rate problems often involve finding how much of a job gets done in a certain amount of time.
These problems can be solved by identifying how quickly a task is completed (rate), and then using that rate to calculate portions of the job done over different periods.
In our exercise, we found the grader’s rate as \( \frac{1}{m} \) jobs per hour. This means that in one hour, \( \frac{1}{m} \) of the job is completed. To find out how much of the job the grader completes in 2 hours, we use this rate.

Multiplication of Fractions

In rate problems, we often have to multiply fractions to find the solution. For our problem, once we know the rate is \( \frac{1}{m} \), we need to find how much work that rate accomplishes in 2 hours. This requires multiplying the rate by 2: \[ \text{Job completed} = \frac{1}{m} \times 2 = \frac{2}{m} \]
When multiplying fractions, multiply the numerators together and the denominators together. In this case, we multiply 1 (the numerator of \( \frac{1}{m} \)) by 2, which gives 2, and \( m \) (the denominator of \( \frac{1}{m} \)) remains the same.
So, the grader completes \( \frac{2}{m} \) of the job in 2 hours.