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.
