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It takes Jack 2 more hours than Tom to type 20 pages. Working together, Jack and Tom can type 25 pages in 3 hours. How long will it take Jack to type 40 pages?
This problem was also posted in PS subforum. Below is my solution from there.
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
Let the time needed for Jack to type 20 pages by \(j\) hours, then for Tom it would be \(j-2\) hours. So the rate of Jack is \(rate=\frac{job}{time}=\frac{20}{j}\) pages per hour and the rate of Tom \(rate=\frac{job}{time}=\frac{20}{j-2}\) 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 --> \(\frac{20}{j}+\frac{20}{j-2}=\frac{25}{3}\) --> \(\frac{60}{j}+\frac{60}{j-2}=25\). 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: \(2j=12\) --> \(j=6\) --> \(\frac{60}{6}+\frac{60}{6-2}=10+15=25\).
first step cross multiplication . For 25 pages -------------- 3 hours Hence for 20 pages --------------- 20*3/25 = 60/25 hrs.
Now let tom alone take t hours for 20 pages as the question states jack takes t+2 hours
Now formula says 1/t + 1/(t+2) = 25/60 Solving we get t= 4 hours. So tom takes 4 hours to print 20 pages. ==> Jack takes 6 hours Definitely he has to take double the time for 40 pages. Hence answer is 12 hours.
Can anyone explain why the following approach comes close, but doesn't exactly match the answer?
Jack spends 2 more hours than Tom to type 20 pages. Jack and Tom together spend 3 hours to type 25 pages. To equate both statements, we can adjust the 25 pages over 3 hours to 20 pages over 2.4 hours (20% decrease on both pages and time).
Knowing that it takes 2.4 hours for two people to complete a task, and assuming that each works at par for simplicity, you know that each would have take 4.8 hours independently to yield 2.4 hours together (1/4.8 + 1/4.8 = 1/2.4)
Since we know that they aren't at par with Jack taking 2 more hours, you can adjust the 4.8 hours of individual work to 3.8 and 5.8 for Tom and Jack respetively. This means it takes 5.8 hours alone for Jack to type 20 pages. Multiply this rate by 2 for 40 pages and we get 11.6, which is close to 12, but not 12.
What did I do wrong?
You did nothing wrong. You approximated the solution, and got a good enough approximated value. Although on some exercises this may not work.
First statement ==> rate jack + rate Tom = 25pages/ 3h ==> 1/J + 1/T = 25/3 Second statement ==> 1/J= 20/(x+2) AND 1/T=20/x substitute in first statement ==> 20/(x+2) + 20/x = 25/3 simplify by 5 and add up on the left to (8x+8)/(x^2+2x)=5/3 cross multiply ==> 24x+24=5x^2+10x 2nd degree equation ==> 5x^2-14x-24=0 Use formula x= (-b +- (b^2-4ac)^0,5)/2a x= (14 +- (14^2+4*5*24)^0,5)/10 14^2+4*5*24=2^2*(7^2+120)=2^2*169=2^2*13^2 so x= (14+-26)/10 only positive x ==> x=4 tom needs 4 h for 20 pages jack needs 4+2=6h for 20 pages. ==> jack needs 12h for 40 pages!
Question is what is the value of 2t+4 ? We know 2t+4 is one among 5,6,8,10,12
Hit and Trial
if 2t+4 = 5 then t = 0.5 does not satisfy equation
if 2t+4 = 6 then t = 2 does not satisfy equation
if 2t+4=12 then t=12 satisfy the equation got the answer.
Please let me know your views with this approach.
Approach is correct, math is not: If 2t+4 = 6 then t = 1, not 2; If 2t+4 = 12 then t = 4, which satisfies 20/t + 20/(t+2) = 25/3 (where t is the time needed for Tom to type 20 pages).
If you refer to my solution above, you'll see that it's basically the same as yours except I took j to be the time needed for Jack to type 20 pages. As we are asked about the time needed for Jack to type 40 pages then this notation will simplify a little bit the final stage of calculation.
This problem was also posted in PS subforum. Below is my solution from there.
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
Let the time needed for Jack to type 20 pages by \(j\) hours, then for Tom it would be \(j-2\) hours. So the rate of Jack is \(rate=\frac{job}{time}=\frac{20}{j}\) pages per hour and the rate of Tom \(rate=\frac{job}{time}=\frac{20}{j-2}\) 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 --> \(\frac{20}{j}+\frac{20}{j-2}=\frac{25}{3}\) --> \(\frac{60}{j}+\frac{60}{j-2}=25\). 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: \(2j=12\) --> \(j=6\) --> \(\frac{60}{6}+\frac{60}{6-2}=10+15=25\).
Can anyone explain why the following approach comes close, but doesn't exactly match the answer?
Jack spends 2 more hours than Tom to type 20 pages. Jack and Tom together spend 3 hours to type 25 pages. To equate both statements, we can adjust the 25 pages over 3 hours to 20 pages over 2.4 hours (20% decrease on both pages and time).
Knowing that it takes 2.4 hours for two people to complete a task, and assuming that each works at par for simplicity, you know that each would have take 4.8 hours independently to yield 2.4 hours together (1/4.8 + 1/4.8 = 1/2.4)
Since we know that they aren't at par with Jack taking 2 more hours, you can adjust the 4.8 hours of individual work to 3.8 and 5.8 for Tom and Jack respetively. This means it takes 5.8 hours alone for Jack to type 20 pages. Multiply this rate by 2 for 40 pages and we get 11.6, which is close to 12, but not 12.
Is there an easier way to get the answer? I can get the formula. But when I find the root it really takes too much time. How do i solve this in 2 min? Should I back solve by dividing all the answers by 2?
Ok so I must need some serious help. I get all the way to 1/t + 1/(t+2) = 25/60 but then I'm not seeing how everyone is solving this to get t=4 so quickly. Help please.
Hellooo ... Can somobody exolain how to solve -5T^2 + 14T + 24 = 0 ? or similar equations? (not the simple ones, they are easily solvable) with quadratic formula? or any shortcut? thanks
together in one hour they can do (20/t + 20/t+2) pages
and by the second statement that is equal to 25/3
solving we get t = 4 hours.
john can type 20 pages in 6 hours and forty pages in 12 hours
I got the equation but how the heck do you guys solve it within 3 minutes?
There has to be a faster way. Can someone please enlighten me? _________________
I'm trying to not just answer the problem but to explain how I came up with my answer. If I am incorrect or you have a better method please PM me your thoughts. Thanks!