It takes Jack 2 more hours than Tom to type 20 pages. If

Bunuel, can you explain how below is possible? Combined 20 pg and equaled to 25 pg?

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 --> \(\frac{20}{j}+\frac{20}{j-2}=\frac{25}{3}\)

We are told that "working together, Jack and Tom can type 25 pages in 3 hours", thus their combined rate is 25/3 pages per hour.

We also know that Jack's rate is 20/j pages per hour and Tom's rate is 20/(j-2) pages per hour, thus their combined rate is 20/j+20/(j-2) pages per hour.

So, we have that 20/j+20/(j-2)=25/3.

Hope it's clear.

I was not able to solve the quadratic; 20/j + 20/j+2 = 25/3
I ended up simplifying it to 5x^2 - 14x - 24 = 0 ; I got the value of x as 4 mins
and for 40 copies as 8 mins....
can anyone help me , where did i go wrong??????????

You did everything right except the last part of assigning the right variable to the right person.

The quadratic you got was 20/j + 20/j+2 = 25/3
Since Jack takes 2 more hrs than Tom, 20/j is the rate of Tom and 20/(j+2) is the rate of Jack.

You got j = 4
So j+2 = 6 = Time taken by Jack to type 20 pages

So time taken by Jack to type 20 pages = 2*6 = 12

When assigning variables, its always advisable to assign x to the value you need to find i.e. j should have been the time taken by Jack since you need to find Jack's time. Not that it changes the question in any way but you avoid this particular error you committed here.
Also, in that case, it is easier to plug in values from the answer choices and you don't really need to solve the quadratic.

Got stuck in the quadratic. However when i looked a little closely at the question I realized I could work by employing logic.
Since the combined rate of work is 25 pages in 3 hours it implies their combined rate is just marginally greater than 8 pages per hour. The question requires us to determine how long Jack takes working alone to type 40 pages. Now since their combined rate is 8 pages an hour working together they would finish 40 pages in about 5 hours. We also know that Jack's rate is slower than Tom's. So if Jack works alone he must take more than double the time it would have taken had they worked together. i.e little more than 10. Only E suffices.

P.S. Could anyone let me know if we should follow this approach of using Logic to arrive at the answer if the Work Rate problems become convoluted??(Like involving massive quadratics)

Yes, your approach is absolutely fine. If nothing is working out, you can certainly take a very good guess in this way (here the options are such that you would have no doubt that (E) is the correct answer).
We just run the risk of more than 1 options satisfying the constraints we come up with at the end. Say, an option gave you 11.6, then we would have wasted time. We cannot use approximation in that case. Therefore, you also need to learn to work on equations where the variable is in the denominator. You don't necessarily have to make a quadratic in that case. You can plug in the options as I have discussed in my post above: it-takes-jack-2-more-hours-than-tom-to-type-20-pages-if-102407.html#p1024552.

I need some guidance here.. Although I understand the basic work = rate*time formula but when working on problems such as above it doesn't quite strikes me that what is a good way to solve a problem until I end up going through Bunuel or any other expert's post. :l
Such as in this problem, rather than assuming variables for time, i assumed variables for rates and came up with: (j+t)*3 = 25 and 20/j = 20/t + 2 where j and t are respective rate variables for Jack and Tom. And while substituting and solving these equations, I got lost and spent 4 minutes without an answer.
But as soon as I went through Bunuel's and Karishma's posts I noticed that assuming variables for what was asked (i.e. time) would have made the solution easier.

My question is how/what should one practice in order to get into the habit of approaching the problem in a simpler way rather than just coming up with long-routes and unnecessary calculations that might not be even needed for getting the answer.

I know this question is quite subjective, but any help will be appreciated.

Thanks much!

Here are a few tips:
- Many times, you wont need to take any variables. You can use a logical approach though that takes some work and practice first.
- If you do need to take variables, you will usually need to take only one. Let that variable be the one which you need to find. Say, if you need time taken, take the variable T, not rate or work. Very rarely would it be a good idea to take something else as the variable. Also, prefer to work with multiplication and addition rather than division and subtraction i.e. if time take by A is 2 hrs more than time taken by B, assume time taken by B = T and time taken by A = T+2.
- Very rarely would you need to take two variables. If you do, find the relation between the two variables as soon as you can and then bring everything down to a single variable.

With practice, you will know how to identify the right approach quickly. Sometimes, everyone gets lost in a circuitous method - important is to move on and not waste too much time on it.

I did this question in the gmat club test. Couldn't solve it using the traditional approach then used another approach for this.

We know that combined they typed 25 pages in 3 hrs. Hence combined they will type 40 pages in 3*40/25=24/5~=5
I considered that since together they take 5 hrs then individually J's time would be more than 10 hours as his speed his lower than that of Tom.

The only possible answer is E i.e 12 hrs.

Yes, your approach is very good. The only thing I have an issue with is the approximation used.
Their combined time is 4.8 hrs and hence we know that Jack will take more than 9.6 hrs. 10 hrs is a possible candidate for the correct option in that case. Though I would say that if Jack took just a wee bit more than 9.6 hrs, then Tom would have taken a tiny bit less than 9.6 hrs and then the difference in their individual time taken could not be 2 hrs. So yes, (E) must be the answer.

Hello,

why can't it be time taken by tom= T and time taken by Jack= T+2 ? And then the eq becomes \(\frac{20}{T}\) + \(\frac{20}{T+2}\) = \(\frac{25}{3}\)

Please help?

It can be. If you solve it, you will get T = 4. Then, to get time taken by Jack, you just add 2 to it to get 6 hrs.
Note that you can take either variable but it is often better to assume what you have to find (time taken by Jack) since sometimes, we forget the last step. If you directly get the value of J, great. If you first find the value of T, you might forget to add 2 at the end to get the actual answer.

Let Tom takes x hrs and Jack takes x+2 hrs .. so in 1 hr Tom completes 20/x pages and Jack completes 20/(x+2) pages .. Given Both can complete 25 pages in 3 hrs .. so in 1 hr they both can complete 25/3 pages in combination . so 20/x + 20/(x+2) = 25/3 .. on solving we get x = 4 .. so Jack takes 4+2 = 6 hrs to complete 20 pages ..so to complete 40 pages he takes 6*2 = 12 hrs ..

Let Rate of Tom Work = a, Jack work = b. Given, as per question -

\(a*t = b*(t+2) = 20\) -------------1
\((a+b)*3 = 25\), ------------------ 2
By Solving Equation 1 & 2 we get,
\(t = 4 hours\). Thus Jack takes 6 Hours to type 20 pages & 12 Hours to type 40 pages.

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 explain what is wrong with my method
rate of jack =1/x
rate of tom=1/x-2
working together they will take=1/x + 1/x-2 = 1/3
i am not getting the correct answer

How does 1/x + 1/x-2 equal 1/3?

\(\frac{1}{x} + \frac{1}{(x-2)} = \frac{(2x - 2)}{x(x - 2)} = \frac{5}{12}\)

Note how you get 5/12 - The combined rate is 25 pages in 3 hrs. But initially our work done was 20 pages which would then be done in (4/5)*3 = 12/5 hrs. So combined rate is 5/12.

When you solve the above, you get x = 6.

So Jack types 20 pages in 6 hrs. He will type 40 pages in 12 hrs.