M09-05

Official Solution:

Jack takes 2 hours longer than Tom to type 20 pages. When they work together, they can type 25 pages in 3 hours. How long would it take for Jack to type 40 pages by himself?

A. 5 hours
B. 6 hours
C. 8 hours
D. 10 hours
E. 12 hours


Let \(j\) hours represent the time needed for Jack to type 20 pages. For Tom, it would take \(j-2\) hours to type the same 20 pages. Therefore, Jack's rate is given by \(rate = \frac{job}{time}=\frac{20}{j}\) pages per hour, and Tom's rate is \(rate=\frac{job}{time}=\frac{20}{j-2}\) pages per hour.

Their combined rate is the sum of their individual rates, which is \(\frac{20}{j}+\frac{20}{j-2}\) pages per hour. This combined rate is equal to \(\frac{25}{3}\) pages per hour, so we have: \(\frac{20}{j}+\frac{20}{j-2}=\frac{25}{3}\). Multiplying through by 3, we get: \(\frac{60}{j}+\frac{60}{j-2}=25\). At this point, we can either substitute the values from the answer choices or solve the quadratic equation. Since we are asked to find the time needed for Jack to type 40 pages, the answer would be \(2j\) (because \(j\) is the time needed to type 20 pages).

Answer E works:

\(2j=12\);

\(j=6\);

Substituting \(j\) into the equation: \(\frac{60}{6}+\frac{60}{6-2}=10+15=25\).


Answer: E