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Select the correct answer.

Mr. Koger is making copies of the school literary magazine to distribute to students and faculty. He has access to two copy machines: an older machine and a new machine. If he used only the older machine, it would take 80 minutes longer than if he used only the new machine. If he used both machines, he would finish the job in 30 minutes.

When [tex]m[/tex] is the number of minutes it would take to finish the job using only the new machine, the situation is modeled by this rational equation:
[tex]\[
\frac{1}{m} + \frac{1}{m+80} = \frac{1}{30}
\][/tex]

How many minutes would it take to finish the job if Mr. Koger used only the new machine?

A. 20 minutes
B. 40 minutes
C. 50 minutes
D. 60 minutes



Answer :

Let's solve the given problem step-by-step.

We are given the equation:
[tex]\[ \frac{1}{m} + \frac{1}{m + 80} = \frac{1}{30} \][/tex]
where [tex]\( m \)[/tex] is the number of minutes it would take to complete the job using only the new machine.

To find the value of [tex]\( m \)[/tex], we need to solve this equation. Here’s the step-by-step breakdown of the solution:

1. Identify the variables:
- [tex]\( m \)[/tex] represents the time in minutes for the new machine to finish the job alone.
- [tex]\( m + 80 \)[/tex] represents the time in minutes for the old machine to finish the job alone.

2. Set up the equation:
- The job done by the new machine in one minute is [tex]\(\frac{1}{m}\)[/tex].
- The job done by the old machine in one minute is [tex]\(\frac{1}{m + 80}\)[/tex].
- The combined rate of both machines should equal [tex]\(\frac{1}{30}\)[/tex], which corresponds to finishing the job in 30 minutes when both machines are used together.

3. Solve the equation:
- Combine the fractions on the left-hand side:
[tex]\[ \frac{1}{m} + \frac{1}{m + 80} = \frac{1}{30} \][/tex]

- To eliminate the fractions, find a common denominator and set up a single equation to solve for [tex]\( m \)[/tex]. However, rather than going through the algebraic steps, we already have the result from solving this equation:

The solution is:
[tex]\[ m = 40 \][/tex]

Thus, the number of minutes it would take Mr. Koger to finish the job using only the new machine is:

B. 40 minutes.