M17-20

Official Solution:

It takes printer A 4 more minutes than printer B to print 40 pages. If working together, the two printers can print 50 pages in 6 minutes, how long will it take printer A to print 80 pages?

A. 12
B. 18
C. 20
D. 24
E. 30


Let's denote the time taken by printer A to print 40 pages as \(a\) minutes. By the given information, printer B can accomplish the same task in \(a-4\) minutes.

The printing rate of A can be calculated as \(rate = \frac{job}{time} = \frac{40}{a}\) pages per minute, and for B it would be \(rate = \frac{job}{time} = \frac{40}{a-4}\) pages per minute.

The combined printing rate of A and B together is therefore \(\frac{40}{a}+\frac{40}{a-4}\) pages per minute. From the stem we also know that both printers together can print 50 pages in 6 minutes, which means that the combined printing rate of A and B together is \(rate = \frac{job}{time} = \frac{50}{6}\). Hence, \(\frac{40}{a}+\frac{40}{a-4} = \frac{50}{6}\).

Dividing this equation by 40, we obtain \(\frac{1}{a}+\frac{1}{a-4}=\frac{5}{24}\). At this point, we can either substitute values from answer choices or solve the resulting quadratic equation. It's crucial to remember that we're asked for the time printer A would take to print 80 pages. Therefore, our answer should be \(2a\) (since \(a\) is the time required to print 40 pages).

Answer D fits the equation:

If \(2a=24\), then \(a=12\).

Substituting \(a\) into the equation gives \(\frac{1}{12}+\frac{1}{8}=\frac{5}{24}\), which holds true. Hence, the solution is \(2a = 24\).


Answer: D