# Problem of the Day

Renee can mow a lawn in 2 hours, and Jeremy can mow the same lawn in 1 hour and 15 minutes. If Renee starts mowing the lawn at 2:00 P.M., and Jeremy starts helping her at 2:30 P.M., at what time will they finish mowing the lawn, rounded to the nearest minute?

A. 2:35 P.M.

B. 2:50 P.M.

C. 3:05 P.M.

D. 3:25 P.M.

E. 3:55 P.M.

### Solution

[latexpage]

To solve this problem, we need to use the formula $work=rate \cdot time$ or $w=rt$, where ‘work’ measures the number of jobs done. In this case, 1 job will mean ‘mowing one lawn’. First, we will write expressions for their rates in terms of jobs done per hour. Renee can do 1 job every 2 hours. We can write this as, $\displaystyle \frac{1\mathrm{ job}}{2 \mathrm{hours}}$ or, dividing the top and bottom by 2, $\displaystyle \frac{\frac{1}{2}\mathrm{ jobs}}{1\mathrm{hour}}$. Jeremy does 1 job every 1 hour and 15 minutes, or 1 job every $\displaystyle \frac{5}{4}$ hours. His rate, expressed as a simplified fraction isÂ $\displaystyle \frac{\frac{4}{5} \mathrm{jobs}}{1 \mathrm{hour}}$. To get the number of jobs they each do in $t$ hours, you would multiply $t$ by their respective rates. Thus, Renee completes $\displaystyle \frac{1}{2}t$ jobs in $t$ hours and Jeremy completesÂ $\displaystyle \frac{4}{5}t$ jobs in $t$ hours.

We are told that Renee works by herself for half an hour first. Thus, she finished $\displaystyle \frac{1}{4}$ of the job by herself, leaving $\displaystyle \frac{3}{4}$ of the job to do once the two of them start working together. To determine how long they will take to complete the job working together, solve the following equation for $t$:

$\displaystyle \frac{1}{2}t+\frac{4}{5}t=\frac{3}{4}$

The left side of the equation is the sum of the number of jobs they each do in $t$ hours, and the right side of the equation is the number of jobs they have to finish. Solving for $t$, we get $t=\frac{15}{26}$. Remember that $t$ is in hours. To convert it to minutes, multiply by 60: $\displaystyle \frac{15}{26} \cdot 60 \approx 34.6$. So it takes them 35 minutes, rounded to the nearest minute. They started working together at 2:30 P.M., so they will finish at 3:05 P.M.

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