Barkatis wrote:
It takes Jack 2 more hours than Tom to type 20 pages. If working together, Jack and Tom can type 25 pages in 3 hours, how long will it take Jack to type 40 pages?
A. 5
B. 6
C. 8
D. 10
E. 12
We can let Tom’s rate = 20/x and Jack’s rate = 20/(x+2), and their combined rate is 25/3; thus:
20/x + 20/(x+2) = 25/3
Multiplying by 3x(x+2), we have:
20(3)(x + 2) + 20(3x) = 25(x)(x + 2)
60x + 120 + 60x = 25x^2 + 50x
25^x - 70x - 120 = 0
5x^2 - 14x - 24 = 0
(5x + 6)(x - 4) = 0
x = -5/6 or x = 4
Since x can’t be negative, we see that x must be 4, so Jack's rate is 20/6 = 10/3.
So it takes him 40/(10/3) = 120/10 = 12 hours to type 40 pages.
Alternate Solution:
Let’s pick a common multiple of 20, 25 and 40, such as 200, and calculate the number of hours to type that number of pages for each scenario.
Since Jack takes 2 more hours than Tom to type 20 pages, it will take Jack 20 more hours than Tom to type 200 pages.
Since Jack and Tom working together can type 25 pages in 3 hours, they can type 200 pages in 12 hours.
Let’s denote the number of hours for Tom to type 200 pages by t. Then, in one hour, Tom can complete 1/t of the job. Since Jack takes 20 more hours than Tom to do the same job, it will take him t + 20 hours to type 200 pages and he can complete 1/(t + 20) of the job in one hour. Working together, they can complete the same job (200 pages) in 24 hours; thus in one hour, they complete 1/24 of the job. We can create the following equation:
1/t + 1/(t + 20) = 1/24
Let’s mutliply each side by 24t(t + 20):
24(t + 20) + 24t = t(t + 20)
24t + 480 + 24t = t^2 + 20t
t^2 - 28t - 480 = 0
(t - 40)(t + 12) = 0
t = 40 or t = -12
Since t cannot be negative, t must equal 40. Since it takes Tom 40 hours to type 200 pages, it will take him 1/5 the number of hours to type 1/5 the number of pages (40 pages); thus Tom will type 40 pages in 40 x 1/5 = 12 hours.
Answer: E
Can you please elaborate on how you solved the quadratic equation?