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## Q. 1.6.3

Solving a Work Rate Problem

One printer can do a job twice as fast as another. Working together, both printers can do the job in 2 hr. How long would it take each printer, working alone, to do the job?

## Verified Solution

Step 1   Read the problem. We must find the time it would take each printer, working alone, to do the job.

Step 2   Assign a variable. Let x represent the number of hours it would take the faster printer, working alone, to do the job. The time for the slower printer to do the job alone is then 2x hours.

Therefore, $\frac{1}{x}$ = the rate of the faster printer (job per hour)

and            $\frac{1}{2x}$ = the rate of the slower printer (job per hour).

The time for the printers to do the job together is 2 hr. Multiplying each rate by the time will give the fractional part of the job completed by each.

 Rate Time Part of the Job Completed $\begin{matrix} _{\nwarrow } \\ ^{\swarrow} \end{matrix}$ A = rt Faster Printer $\frac{1}{x}$ 2 $2\left(\frac{1}{x}\right)=\frac{2}{x}$ Slower Printer $\frac{1}{2x}$ 2 $2\left(\frac{1}{2x}\right)=\frac{1}{x}$

Step 3 Write an equation. The sum of the two parts of the job completed is 1 because one whole job is done.

$\overset{\begin{matrix} \text{Part of the job} \\ \text{done by the}\\\underbrace{\text{faster printer }}_{} \end{matrix} }{\frac{2}{x} } \overset{\begin{matrix} \\ +\\ \end{matrix} }{+} \overset{\begin{matrix} \text{Part of the job} \\\text{done by the}\\ \underbrace{\text{slower printer} }_{} \end{matrix} }{\frac{1}{x} } \overset{\begin{matrix} \\= \\ \end{matrix} }{=} \overset{\begin{matrix} \\\text{One whole} \\ \underbrace{\text{ job }}_{} \end{matrix} }{1}$

Step 4   Solve.  $x\left(\frac{ 2 }{x} + \frac{1}{ x} \right) = x(1)$          Multiply each side by

x, where x ≠ 0.

$x\left(\frac{ 2}{x}\right) + x\left(\frac{1}{ x}\right) = x(1)$       Distributive property

2 + 1 = x                                   Multiply.