Printer a takes 40 more minutes to print the pages of a book

\(\frac{1}{T+40}+\frac{1}{T+10}=\frac{1}{T}\)

\(T=20\)

\(\frac{a}{b}=\frac{60}{30}=\frac{2}{1}\)

FIRST OF ALL:

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AS FOR THE QUESTION:
Let the time needed for printer A be \(a\) minutes and the time needed for printer B be \(b\) minutes. Then the time needed for printers A and B working together will be \(\frac{ab}{a+b}\) minutes.

Given: \(a-40=\frac{ab}{a+b}\) and also \(b-10=\frac{ab}{a+b}\). Question: \(\frac{a}{b}=?\)

\(a-40=b-10\) --> \(b=a-30\);

\(a-40=\frac{ab}{a+b}\) --> \(a^2-40a-40b=0\), as \(b=a-30\) then \(a^2-80a+1,200=0\) --> \(a=20\) (not a valid solution as in this case \(b\) becomes negative) or \(a=60\) --> \(b=a-30=30\) --> \(\frac{a}{b}=\frac{2}{1}=\frac{2}{1}\).

Answer: B.

Check this for theory on work problems:
word-translations-rates-work-104208.html?hilit=printer#p812628

Hope it helps.
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is it so obvious ?
x+40/x+10 = 4/1

Bunuel has already given the algebraic approach.
Now let's try to solve it logically.

Lets say they work for x minutes together and complete the work.
a, working alone, takes 40 mins extra - work done by A is 40 mins is the amount of work that was done by b in x minutes when they worked together.
So time taken by a: time taken by b = 40:x .... (I)
Similarly, when b works alone, he takes 10 mins extra - the same work was done by a in x mins when they worked together
time taken by a:time taken by b = x:10 ......(II)
From (I) and (II), 40/x = x/10 or x = 20
So time taken by a: time taken by b = 40:x = 40:20 = 2:1
Answer (B)
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I tried to pick a number but that's wrong, why?

a+b complete the job in 10 minutes
a in 10+40, b in 10 + 10

This is wrong on many levels.

You cannot pick some random numbers and hope that they will give you the required ratio. Do your numbers satisfy the stem saying that "printer a takes 40 more minutes to print the pages of a book than do the printers a and b together, whereas printer b takes 10 more minutes to print the same pages than do the printers a and b together"?

Also, you made the same mistake as you did here: two-water-pumps-working-simultaneously-at-their-respective-155865.html#p1270965

Seems that you need to brush up fundamentals. Check here: machines-x-and-v-produced-identical-bottles-at-different-104208.html#p812628

Hope it helps.
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I didn't understand how we concluded this >>> So Speed of a: speed of b = 40:x .... (I)
Can you please shed some light...

Yeah, I'm trying to
Thanks!

This method makes use of ratios. I have discussed the use of ratios in work problems here: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/03 ... -problems/

Imagine the case in which both printers are working together and printing, say 100 pages in x mins. Let's say printer a prints 40 pages and printer b prints 60 pages in this time.

Say, you get another 100 pages to print but you notice that printer b is busy. So you use only printer a to print the 100 pages. This time it takes you more time (obviously) - it takes you (x + 40) mins.
What can we conclude? We can say that printer a takes 40 mins to do the work that printer b was doing in x mins. Printer b was printing those 60 pages in x mins but now printer a has to do that work too and it takes an extra 40 mins to print those pages printed by b previously.

Time taken by printer a to do some work : time taken by printer b to do the same work = 40:x (Actually, I am working with time taken all through but writing speed. Will edit it)
Similarly, Time taken by printer a to do some work:time taken by printer b to do the same work = x:10
x = 20
Time taken by printer a to do some work : time taken by printer b to do the same work = 40:20 = 2:1
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Another approach:

A: t+40
A and B : t
B: t+10

Equation Rate(A and B) = Rate(A) + Rate(B)

1/(t+40) + 1/(t+10) = 1/t

2t^2 +50t = t^2+50t+400

t^2=400
t=20

Ratio is t+40/(t+20) hence (B)

can you please explain this part """"Lets say they work for x minutes together and complete the work.
a, working alone, takes 40 mins extra - work done by A is 40 mins is the amount of work that was done by b in x minutes when they worked together.
So time taken by a: time taken by b = 40:x .... (I)"""""
once again , i can not come to terms with the solution ( how can a:b equals 40 : x)

Take another example:

Say you are your friend work together on a project (say, writing 8 pages) and complete it in 4 hours (both write one page each hour). If you work alone, you take 8 hrs to complete it (one page each hour). Why? Because you need extra 4 hrs to complete the work (4 pages) which was previously done by your friend. Here, the numbers assumed that you and your friend have the same speed of work and hence did equal work when you worked together.

Now imagine this, you are your friend work together on a project (8 pages) and complete it in 4 hours but if you work alone, you take 7 hrs to complete it? Why? Because you need extra 3 hrs to complete the work which was previously done by your friend. Now, we see that you are faster than your friend. In 3 hrs, you complete what he was supposed to do in 4 hrs (while you were also working). So you write more than 1 page an hour while your friend writes less than one page an hour. The work for which he takes 4 hrs, you take only 3 hrs.

Time taken by him:Time taken by you = 4:3

Does this make sense?

If yes, then come back to this question. It is the same concept.

Both take x mins together.
a takes 40 mins extra when working alone. This means that in 40 mins, a does the work b was supposed to complete in x mins.

Time taken by a:Time taken by b = 40:x
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Let the time for both printers working together to print the pages be x. We see that printer a needs x + 40 minutes and printer b needs x + 10 minutes to print these pages.

Since rate = work/time, printer a completes 1/(x + 40) of the job per minute and printer b completes 1/(x + 10) of the job per minute. Together, they complete:

1/(x + 40) + 1/(x + 10) = (2x + 50)/[(x + 40)(x + 10)] of the job per minute

Since we know that they need x minutes to complete the job, the fraction of the job they can complete per minute (when they work together) is also equal to 1/x. We can equate the two expressions and solve for x:

(2x + 50)/[(x + 40)(x + 10)] = 1/x

2x^2 + 50x = (x + 40)(x + 10)

2x^2 + 50x = x^2 + 50x + 400

x^2 = 400

x = 20

So, it takes 20 minutes to get the job done when both printers work together. From the given information, printer a needs 20 + 40 = 60 minutes and printer b needs 20 + 10 = 30 minutes. The ratio of the time of printer a to that of printer b is 60:30 which is equivalent to 2:1.

Answer: B
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My solution was the very similar to JeffTargetTestPrep;


Equation:

"x" denotes "total time"

Step1:

\(\frac{1}{x+40} + \frac{1}{x+10} = \frac{1}{x}\)


Step 2: Expand

\(\frac{x+10+x+40}{x^2+10x+40x+400}=\frac{1}{x}\)


Step 3: Simplify

\(\frac{2x+50}{x^2+50x+400}=\frac{1}{x}\)

\(2x+50=\frac{x^2+50x+400}{x}\)

\(2x+50=x+50+\frac{400}{x}\)

\(x=\frac{400}{x}\)

\(x^2=400\)

\(x=20\)


Step 4: Put "x" (20) back into the original equation:

\(\frac{1}{20+40} + \frac{1}{20+10} = \frac{1}{20}\)


Final equation:

\(\frac{1}{60} +\frac{1}{30}= \frac{1}{20}\)

As you can see, the ratio for machine A to B is 60:30 or 2:1

1. Let the time taken when both print together be x.
2. Time taken by A is x+40 and time taken by B is x+10
3. 1/(x+40) +1/(x+10) = 1/x.
4. x=20, So the ratio is 2:1.
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Answer: option B


Check solution attached

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