Official Solution:

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


Let the time needed for Jack to type 20 pages be \(j\) hours, then for Tom it would be \(j-2\) hours. So the rate of Jack is \(\text{rate}=\frac{\text{job}}{\text{time}}=\frac{20}{j}\) pages per hour and the rate of Tom \(\text{rate}=\frac{\text{job}}{\text{time}}=\frac{20}{j-2}\) pages per hour.

Their combined rate would be \(\frac{20}{j}+\frac{20}{j-2}\) pages per hour and this equal to \(\frac{25}{3}\) pages per hour, so: \(\frac{20}{j}+\frac{20}{j-2}=\frac{25}{3}\), Multipy by 3: \(\frac{60}{j}+\frac{60}{j-2}=25\). At this point we can either try to substitute the values from the answer choices or solve quadratic equation. Remember as we are asked to find time needed for Jack to type \(40\) pages, then the answer would be \(2j\) (as \(j\) is the time needed to type \(20\) pages).

Answer E works:

\(2j=12\);

\(j=6\);

\(\frac{60}{6}+\frac{60}{6-2}=10+15=25\)
.

Answer: E
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Hi Bunuel I solved this problem a different way, but the result is different plz correct me why I am wrong !
Jack: x ( page/hr)
Tom: y ( page/hr)
we have : x + y = 25/3 (page/hr); (1/x - 1/y) = (2/20)
solving these two equations I have 2 values of x, which are x1 = 25 (p/hr) and x2 = 10/3 (p/hr)
so I cannot figure out the expected result !!!!
tks Bunuel

If x = 25, then y comes out to be negative, so this is not a valid solution.
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but where am I wrong with this approach ?

Your approach is fine. At the rate of 10/3 pages per hour it would take Jack 40/(10/3) = 12 hours to type 40 pages.
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I really like the way you present the work equations Bunuel (similar to Speed/Distance/Time)...it helps simplify the equations immensely...thx for the tip...

Can you please do it by the quadratic equation ? Because I don't recall these short cuts in exam so I have to go through quadratic equation. But I am unable to solve this through quadratic equation.

Check alternative approaches to this question here: it-takes-jack-2-more-hours-than-tom-to-type-20-pages-if-102407.html

Hope it helps.
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Hi Bunuel, there's a typo in the explanation. I believe "by" should be "be".

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Edited. Thank you.
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I received a PM about this problem.

Since Jack and Tom together can type 25 pages in 3 hours, their combined rate = \(\frac{work}{time}\) = \(\frac{25}{3}\) = 8\(\frac{1}{3}\) pages per hour.

We can PLUG IN THE ANSWERS, which represent Tom's time to type 40 pages.
When the correct answer choice is plugged in, the combined rate for Jack and Tom will be 8\(\frac{1}{3}\) pages per hour.

D: 10 hours
Here, Jack takes 10 hours to type 40 pages, implying that Jack's time to type 20 pages = 5 hours.
Thus, Jack's rate = \(\frac{work}{time}\) = \(\frac{20}{5}\) = 4 pages per hour.

Since Jack takes 2 more hours than Tom to type 20 pages, Tom's time to type 20 pages = 5-2 = 3 hours.
Thus, Tom's rate = \(\frac{work}{time}\) = \(\frac{20}{3}\) = 6\(\frac{2}{3}\) pages per hour.

Combined rate for Jack and Tom = 4 + 6\(\frac{2}{3}\) = 10\(\frac{2}{3}\) pages per hour.
Their combined rate is TOO FAST.
To decrease their combined rate, Jack must take MORE TIME to type 40 pages.

The correct answer is: .
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Bunuel
In your solution you assumed the time taken by jack is j hours and the time taken by Tom is j-2 hours and you built the equation:
20/j +20/j-2 = 25/3. and you got j= 6.

After i read the question: i assumed the time taken by Tom is t hours and that taken by jack is t+2 hour (since jack took 2 more hours than tom) and my equation is
20/t+2 + 20/t = 25/3. and i got t = 4. and 2t = 8.

Can you please point what is wrong with my way of building the equation.

Thank you

Nothing wrong. If t = 4, then j = 6 - the same answer as in my solution. Notice that the question asks: how long will it take Jack to type 40 pages?
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I don't agree with the explanation. If Tom takes j hours , then Jack should take J+2 hours... If I am solving like this my answer is not coming.

Can you please elaborate why this approach is wrong?

The solution via the quadratic is as follows.

\(\frac{4}{(t+2)}+\frac{4}{t}=\frac{5}{3} \implies 0=5t^2-14t-24 \implies 0=(t-4)(5t+6) \implies t=4 \implies \frac{20}{6}*x=40 \implies x=12\)
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