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M17-20

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Math Expert
Joined: 02 Sep 2009
Posts: 52144

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16 Sep 2014, 00:01
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Difficulty:

95% (hard)

Question Stats:

53% (02:33) correct 47% (03:09) wrong based on 220 sessions

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It takes printer A 4 more minutes than printer B to print 40 pages. If working together, the two printers can print 50 pages in 6 minutes, how long will it take printer A to print 80 pages?

A. 12
B. 18
C. 20
D. 24
E. 30

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Joined: 02 Sep 2009
Posts: 52144

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16 Sep 2014, 00:01
Official Solution:

It takes printer A 4 more minutes than printer B to print 40 pages. If working together, the two printers can print 50 pages in 6 minutes, how long will it take printer A to print 80 pages?

A. 12
B. 18
C. 20
D. 24
E. 30

Let the time needed to print 40 pages for printer A be $$a$$ minutes, so for printer B it would be $$a-4$$ minutes.

The rate of A would be $$rate=\frac{job}{time}=\frac{40}{a}$$ pages per minute and the rate of B $$rate=\frac{job}{time}=\frac{40}{a-4}$$ pages per minute.

Their combined rate would be $$\frac{40}{a}+\frac{40}{a-4}$$ pages per minute. Also as "two printers can print 50 pages in 6 minutes" then their combined rate is $$rate=\frac{job}{time}=\frac{50}{6}$$, so $$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$.

$$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$. Divide by 40: $$\frac{1}{a}+\frac{1}{a-4}=\frac{5}{24}$$. At this point we can either try to substitute the values from answer choices or solve quadratic equation. Remember as we are asked to find the time needed for printer A to print $$80$$ pages, then the answer would be $$2a$$ (as $$a$$ is the time needed to print $$40$$ pages).

$$2a=24$$;

$$a=12$$;

$$\frac{1}{12}+\frac{1}{8}=\frac{5}{24}$$.

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22 Mar 2016, 21:08
Bunuel wrote:
Official Solution:

It takes printer A 4 more minutes than printer B to print 40 pages. If working together, the two printers can print 50 pages in 6 minutes, how long will it take printer A to print 80 pages?

A. 12
B. 18
C. 20
D. 24
E. 30

Let the time needed to print 40 pages for printer A be $$a$$ minutes, so for printer B it would be $$a-4$$ minutes.

The rate of A would be $$rate=\frac{job}{time}=\frac{40}{a}$$ pages per minute and the rate of B $$rate=\frac{job}{time}=\frac{40}{a-4}$$ pages per minute.

Their combined rate would be $$\frac{40}{a}+\frac{40}{a-4}$$ pages per minute. Also as "two printers can print 50 pages in 6 minutes" then their combined rate is $$rate=\frac{job}{time}=\frac{50}{6}$$, so $$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$.

$$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$. Divide by 40: $$\frac{1}{a}+\frac{1}{a-4}=\frac{5}{24}$$. At this point we can either try to substitute the values from answer choices or solve quadratic equation. Remember as we are asked to find the time needed for printer A to print $$80$$ pages, then the answer would be $$2a$$ (as $$a$$ is the time needed to print $$40$$ pages).

$$2a=24$$;

$$a=12$$;

$$\frac{1}{12}+\frac{1}{8}=\frac{5}{24}$$.

I tried to solve this question by considering time taken by B as "a" and that by printer A as "a+4" but was unable to solve the question. Why should B's time be "a" and A's time be "a-4" and not the other way around?
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Joined: 02 Aug 2009
Posts: 7200

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22 Mar 2016, 21:16
vdhaval wrote:
Bunuel wrote:
Official Solution:

It takes printer A 4 more minutes than printer B to print 40 pages. If working together, the two printers can print 50 pages in 6 minutes, how long will it take printer A to print 80 pages?

A. 12
B. 18
C. 20
D. 24
E. 30

Let the time needed to print 40 pages for printer A be $$a$$ minutes, so for printer B it would be $$a-4$$ minutes.

The rate of A would be $$rate=\frac{job}{time}=\frac{40}{a}$$ pages per minute and the rate of B $$rate=\frac{job}{time}=\frac{40}{a-4}$$ pages per minute.

Their combined rate would be $$\frac{40}{a}+\frac{40}{a-4}$$ pages per minute. Also as "two printers can print 50 pages in 6 minutes" then their combined rate is $$rate=\frac{job}{time}=\frac{50}{6}$$, so $$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$.

$$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$. Divide by 40: $$\frac{1}{a}+\frac{1}{a-4}=\frac{5}{24}$$. At this point we can either try to substitute the values from answer choices or solve quadratic equation. Remember as we are asked to find the time needed for printer A to print $$80$$ pages, then the answer would be $$2a$$ (as $$a$$ is the time needed to print $$40$$ pages).

$$2a=24$$;

$$a=12$$;

$$\frac{1}{12}+\frac{1}{8}=\frac{5}{24}$$.

I tried to solve this question by considering time taken by B as "a" and that by printer A as "a+4" but was unable to solve the question. Why should B's time be "a" and A's time be "a-4" and not the other way around?

hi,
printer A= a, B=a-4..
OR
printer B=a and A= a+4..

both the above cases are same and if you have taken the second case, be careful in that you are getting a as value of B and you will have to convert to A's time..
You could be going wrong there, otherwise you should get the answer either way

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04 Aug 2016, 04:15
1
I think this is a high-quality question and I agree with explanation.
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26 Aug 2016, 02:18
Bunuel wrote:
Official Solution:

It takes printer A 4 more minutes than printer B to print 40 pages. If working together, the two printers can print 50 pages in 6 minutes, how long will it take printer A to print 80 pages?

A. 12
B. 18
C. 20
D. 24
E. 30

Let the time needed to print 40 pages for printer A be $$a$$ minutes, so for printer B it would be $$a-4$$ minutes.

The rate of A would be $$rate=\frac{job}{time}=\frac{40}{a}$$ pages per minute and the rate of B $$rate=\frac{job}{time}=\frac{40}{a-4}$$ pages per minute.

Their combined rate would be $$\frac{40}{a}+\frac{40}{a-4}$$ pages per minute. Also as "two printers can print 50 pages in 6 minutes" then their combined rate is $$rate=\frac{job}{time}=\frac{50}{6}$$, so $$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$.

$$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$. Divide by 40: $$\frac{1}{a}+\frac{1}{a-4}=\frac{5}{24}$$. At this point we can either try to substitute the values from answer choices or solve quadratic equation. Remember as we are asked to find the time needed for printer A to print $$80$$ pages, then the answer would be $$2a$$ (as $$a$$ is the time needed to print $$40$$ pages).

$$2a=24$$;

$$a=12$$;

$$\frac{1}{12}+\frac{1}{8}=\frac{5}{24}$$.

why do we have to take 40/a+40/a−4 and not 40/a+40/a+4 why is my answer not coming right?
Math Expert
Joined: 02 Sep 2009
Posts: 52144

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26 Aug 2016, 03:01
Bunuel wrote:
Official Solution:

It takes printer A 4 more minutes than printer B to print 40 pages. If working together, the two printers can print 50 pages in 6 minutes, how long will it take printer A to print 80 pages?

A. 12
B. 18
C. 20
D. 24
E. 30

Let the time needed to print 40 pages for printer A be $$a$$ minutes, so for printer B it would be $$a-4$$ minutes.

The rate of A would be $$rate=\frac{job}{time}=\frac{40}{a}$$ pages per minute and the rate of B $$rate=\frac{job}{time}=\frac{40}{a-4}$$ pages per minute.

Their combined rate would be $$\frac{40}{a}+\frac{40}{a-4}$$ pages per minute. Also as "two printers can print 50 pages in 6 minutes" then their combined rate is $$rate=\frac{job}{time}=\frac{50}{6}$$, so $$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$.

$$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$. Divide by 40: $$\frac{1}{a}+\frac{1}{a-4}=\frac{5}{24}$$. At this point we can either try to substitute the values from answer choices or solve quadratic equation. Remember as we are asked to find the time needed for printer A to print $$80$$ pages, then the answer would be $$2a$$ (as $$a$$ is the time needed to print $$40$$ pages).

$$2a=24$$;

$$a=12$$;

$$\frac{1}{12}+\frac{1}{8}=\frac{5}{24}$$.

why do we have to take 40/a+40/a−4 and not 40/a+40/a+4 why is my answer not coming right?

Please check the discussion above. For example, this post: m17-184126.html#p1663300
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Joined: 01 Oct 2014
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26 Aug 2016, 03:06
Bunuel wrote:
Bunuel wrote:
Official Solution:

It takes printer A 4 more minutes than printer B to print 40 pages. If working together, the two printers can print 50 pages in 6 minutes, how long will it take printer A to print 80 pages?

A. 12
B. 18
C. 20
D. 24
E. 30

Let the time needed to print 40 pages for printer A be $$a$$ minutes, so for printer B it would be $$a-4$$ minutes.

The rate of A would be $$rate=\frac{job}{time}=\frac{40}{a}$$ pages per minute and the rate of B $$rate=\frac{job}{time}=\frac{40}{a-4}$$ pages per minute.

Their combined rate would be $$\frac{40}{a}+\frac{40}{a-4}$$ pages per minute. Also as "two printers can print 50 pages in 6 minutes" then their combined rate is $$rate=\frac{job}{time}=\frac{50}{6}$$, so $$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$.

$$\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}$$. Divide by 40: $$\frac{1}{a}+\frac{1}{a-4}=\frac{5}{24}$$. At this point we can either try to substitute the values from answer choices or solve quadratic equation. Remember as we are asked to find the time needed for printer A to print $$80$$ pages, then the answer would be $$2a$$ (as $$a$$ is the time needed to print $$40$$ pages).

$$2a=24$$;

$$a=12$$;

$$\frac{1}{12}+\frac{1}{8}=\frac{5}{24}$$.

why do we have to take 40/a+40/a−4 and not 40/a+40/a+4 why is my answer not coming right?

Please check the discussion above. For example, this post: m17-184126.html#p1663300

Yeah got it,Thanks a lot..! lol ,major silly mistake!
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22 Aug 2017, 12:13
Bunuel wrote:
It takes printer A 4 more minutes than printer B to print 40 pages. If working together, the two printers can print 50 pages in 6 minutes, how long will it take printer A to print 80 pages?

A. 12
B. 18
C. 20
D. 24
E. 30

I like the solution when after we get to 1/a+1/(a−4)=5/24, we check the answer choices.

I solved the full quadr equation:
50/ (40/(t+4) + 40/t) = 6 - got pretty big numbers and although i solved it correct, it took me over 4 minutes, which is unacceptable. (We have to find 2t+8 in this case).

But then I realized, that we can simplify this equation.
It takes printer A 4 more minutes than printer B to print 40 pages = It takes printer A 1 more minute than printer B to print 10 pages
In this case we have to solve next equation:
50/(10/(t+1) + 10/t) = 6 (and to find 2t+8)- this is much faster than the first one.
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GPA: 3.8

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09 Nov 2017, 12:20
1
Speaking of taking 4 minutes...

Found it quicker to solve this in terms of printer B:

Time it takes printer A to do 40 pages, $$A[40], = B+4$$ minutes
Thus, to do 80 pages as question asks, $$A[80], = 2(B+4)$$ minutes

Next, $$R$$A $$+$$ $$R$$B $$=$$ $$\frac{25}{3}$$ $$=>$$ $$\frac{40}{B+4}$$ + $$\frac{40}{B}$$ = $$\frac{25}{3}$$

At this point, eyeballing is useful: $$B=10$$ or $$B=5$$ are clearly out and a little mental stretching suggests only $$B=8$$ fits the quadratic

If $$B=8$$, then $$A[80] = 2(8+4) = 24$$ minutes
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Joined: 31 Oct 2016
Posts: 106

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04 Jan 2018, 01:04
How did I solve using approximation?

Let's assume that both of them are the same printers and let's replace 40 pages by 50 pages. Together they printed 50 pages in 6 minutes. It means that 1 printer can print 50 pages in 12 minutes.
12 minutes - our average time taken by both different printers. Given that A is slower than B for 4 minutes, the only values we can get are 10 and 14 minutes (average is 12, and one is more gradual than another for 4 minutes).

Printer A can print 50 pages in 14 minutes and 100 pages in 28 minutes. Therefore, it can print 80 pages a little bit faster than 28 minutes. Answer - 24 minutes. Confidence - 80%. Time - about 1 minute
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16 Jul 2018, 11:23
Bunuel

Can this be solved using weighted average?
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16 Jul 2018, 11:55
Rates: 40/x
40/(x+4)

40/x*6+40/(x+4)*6=50
240/x+240/(x+4)=50
240(x+4)+240x=50x(x+4)
240x+960+240x=50x^2+200x
0=50x^2-280x-960
0=5x^2-28x-96
0=(5x+12)(x-8)
x=8

40/12 t = 80
t=24 min
Re: M17-20 &nbs [#permalink] 16 Jul 2018, 11:55
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