A new machine that deposits cement for a road requires 13 hours to complete a one-half mile section of road. An older machine requires 15 hours to pave the same amount of road. After depositing cement for 2 hours, the new machine develops a mechanical problem and quits working. The older machine is brought into place and continues the job. How long does it take the older machine to complete the job

To find out how long it takes for the older machine to complete the job, we need to calculate the work rate of each machine first.

Work rate can be calculated using the formula:
work rate = amount of work / time

Let's assume that the work required for paving half a mile of road is represented by "W".

In this case, the new machine takes 13 hours to complete the work, so its work rate is:
work rate of new machine = W / 13

Similarly, the older machine takes 15 hours to complete the same work, so its work rate is:
work rate of older machine = W / 15

Now, let's calculate the work done by the new machine before it develops a mechanical problem.

The new machine works for 2 hours, and since its work rate is W / 13, the work done by the new machine in 2 hours is:
work done by new machine = (W / 13) * 2

After the new machine stops working, the older machine takes over and completes the remaining work, which is (W - work done by new machine).

To find out how long it takes for the older machine to complete the job, we can set up the following equation:

work rate of older machine * time taken by older machine = remaining work

Let's substitute the values into the equation:

(W / 15) * time taken by older machine = (W - work done by new machine)

Simplifying the equation, we get:

time taken by older machine = (W - work done by new machine) / (W / 15)

Now, we need to substitute the value of work done by the new machine:
work done by new machine = (W / 13) * 2

Substituting the value of work done by new machine in the equation, we get:

time taken by older machine = (W - ((W / 13) * 2)) / (W / 15)

Simplifying further:

time taken by older machine = (15W - (2 * (W / 13))) / (W / 15)

Now, we have the expression to calculate the time taken by the older machine to complete the job. However, we don't have a specific value for the work required (W) to pave half a mile of road. Without this information, we cannot find the exact time taken by the older machine to complete the job.