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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,

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by VJesus12 » Thu Jun 04, 2020 7:14 am

Timer

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Answers

A

B

C

D

E

Stats

Difficulty—

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

[spoiler]OA=E[/spoiler]

Source: GMAT Club Tests
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Re: 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

by deloitte247 » Fri Jun 05, 2020 6:40 am
Let hours taken by Tom to type 20 pages = x hours
It will take Jack => x + 2 hours

Rate of work done by Tom in 1 hour = 20/x pages per hour
Rate of work done by Jack in 1 hour = 20/x-2 pages per hour

Working together they can type 25 pages in 3 hours

$$Therefore,\ \frac{20}{x}+\frac{20}{x+2}=\frac{25}{3\ }=>\ in\ 1\ hour$$ $$\frac{20\left(x+2\right)+20\left(x\right)}{x\left(x+2\right)}=\frac{25}{3}$$
$$\frac{20x+40+20x}{x\left(x+2\right)}=\frac{25}{3}$$
$$\frac{40x+40}{x\left(x+2\right)}=\frac{25}{3}$$
$$3\left(40x\ +40\right)=25\left(x^2+2x\right)$$
$$120x+120=25x^2+50x$$
$$0=25x^2+50x-120x-120$$
$$0=25x^2-70x-120$$
$$25x^2-70x-120=0$$
$$5\left(5x^2-14x-24\right)=0$$
$$5\left(5x^2-20x+6x-24\right)=0$$
$$5\left[5x\left(x-4\right)+6\left(x-4\right)\right]=0$$
$$5\left[\left(5x+6\right)\left(x-4\right)\right]=0$$
$$Therefore,\ 5\left(5x+6\right)=0\ or\ 5\left(x-4\right)=0$$
$$=>25x+30=0\ or\ 5x-20=0$$
$$=>25x=-30\ or\ 5x=20$$
$$x=\frac{-30}{25}\ or\ x=\frac{20}{5}$$
$$x=\frac{-6}{5}\ or\ x=4$$
$$Check\ for\ x\ \ in\ \frac{20}{x}+\frac{20}{x+2}=\frac{25}{3}$$
$$If\ x\ =\ 4\ =>$$
$$\frac{20}{4}+\frac{20}{4+2}=\frac{20}{4}+\frac{20}{6}=\frac{60+40}{12}$$
$$\frac{100}{12}=\frac{25}{3}$$
$$If\ x\ =\ \frac{-6}{5}=>$$
$$\left(\frac{20}{\frac{-6}{5}}\right)+\left(\frac{20}{\frac{-6}{5}+\frac{2}{1}}\right)$$
$$=\left(20\cdot\frac{-5}{6}\right)+\left(\frac{20}{\frac{4}{5}}\right)$$
$$=\left(20\cdot\frac{-5}{6}\right)+\left(20\cdot\frac{5}{4}\right)$$
$$=\frac{-100}{6}+\frac{100}{4}$$
$$=\frac{-200+300}{12}=\frac{100}{12}=\frac{25}{3}$$
If x = -6/5 then time taken by Jack to type 20 pages = x + 2
$$i.e\ \frac{-6}{5}+\frac{2}{1}=\frac{-6+10}{5}=\frac{4}{5}$$
Time taken to type 40 pages
$$=\frac{40\cdot\frac{4}{5}}{20}=\frac{32}{20}=\frac{16}{10}=\frac{8}{5}$$
$$=\frac{8}{5}\ hours$$
This answer is wrong, it's not in the option so x cannot be -6/5

If x = 4 then the time taken by Jack to type 20 pages = x + 2 = 4 + 2 = 6 hours
If Jack used 6 hours to type 20 pages, he will use y hours to type 40 pages
$$=>\ y=\frac{6\cdot40}{20}=6\cdot2\ =\ 12\ hours$$

Answer = E
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Re: 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

by Scott@TargetTestPrep » Tue Jun 09, 2020 4:03 am
VJesus12 wrote: ↑
Thu Jun 04, 2020 7:14 am
 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

[spoiler]OA=E[/spoiler]

Solution:

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 multiply 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

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