Question

Two student volunteers are stuffing envelopes for a local food pantry. The mailing will be sent to 560 possible contributors. Luis can stuff 160 envelopes per hour and Mei can stuff 120 envelopes per hour. a. Working alone, what fraction of the job can Luis complete in one hour? in \(t\) hours? Write the fraction in lowest terms. b. Working alone, what fraction of the job can Mei complete in \(t\) hours? c. Write an expression for the fraction of the job that Luis and Mei can complete in \(t\) hours if they work together. d.To find how long it will take Luis and Mei to complete the job if they work together, you can set the expression you wrote in part (c) equal to 1 and solve for \(t\). Explain why this will work. e. How long will it take Luis and Mei to complete the job if they work together? Check your solution.

Short answer

Luis can complete \(\frac{160}{560} = 0.28\) or \(28\%\) of the work per hour and \(0.28t\) fraction in \(t\) hours. Mei can complete \(\frac{120}{560} = 0.21\) or \(21\%\) of the work per hour and \(0.21t\) fraction in \(t\) hours. Working together, the expression for the fraction of job they can complete in \(t\) hours is \(0.28t + 0.21t = 0.49t\). Solving for \(t\) in \(0.49t = 1\) gives \(t \approx 2.04\) hours. Thus, it would take about 2.04 hours for both Luis and Mei to complete the work together. The explanations and results of the check are found in the solutions steps.

Step-by-step solution

Calculate Luis' work rate

First, calculate how much of the job Luis can accomplish within an hour. This calculation is made by dividing the rate at which Luis works (160 envelopes per hour) by the total number of envelopes to be stuffed (560). This calculation would be represented as \(\frac{160}{560}\). The result of this calculation is the fraction of the total work that Luis can complete in one hour. To calculate the work Luis can complete in \(t\) hours, you simply multiply this fraction by \(t\).

Calculate Mei's work rate

Now calculate how much of the job Mei can do per T hours. This is calculated by dividing the rate at which Mei works (120 envelopes per hour) by the total number of envelopes to be stuffed (560). This calculation is represented as \(\frac{120}{560} \times t\). The result of this calculation is the fraction of the total work that Mei can complete in \(t\) hours.

Combine Luis and Mei's work rate

To calculate the fraction of the job that Luis and Mei can complete in \(t\) hours if they work together, you combine their individual rates. This can be represented as \(\frac{160}{560}t + \frac{120}{560}t\). This sum will give the total fraction of the job that they can complete together in \(t\) hours.

Calculate the time required to complete the job

To find how long it will take Luis and Mei to complete the job if they work together, you set the expression from Step 3 equal to 1 and solve for \(t\). This is because 1 represents the whole job, or 100%. Therefore, when the total fraction of the job they can do equals 1, the job is completed. Solving \(\frac{160}{560}t + \frac{120}{560}t = 1\) for \(t\) will give the time it takes for Mei and Luis to complete the job together.

Checking the solution

Substitute this value of \(t\) into the work rates of Luis and Mei separately. Both should give a result less than or equal to 1 (since they can't do more than the whole job). If they give a result more than 1, then there is some error in the calculations. Also, you should check that if you add the work done by Luis and Mei individually (for the calculated \(t\)), it should equal to 1 indicating the whole job is complete. If all these conditions are satisfied, you can have confidence in the result.