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Work/Rate

by BTGmoderatorRO » Wed Dec 13, 2017 5:14 pm
It takes printer A 4 more minutes more than printer B to print 40 pages. 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

OA is D
This question is confusing me.C an expert explain why D is the answer?
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by DavidG@VeritasPrep » Thu Dec 14, 2017 8:37 am
Roland2rule wrote:It takes printer A 4 more minutes more than printer B to print 40 pages. 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

OA is D
This question is confusing me.C an expert explain why D is the answer?
Not a bad one to back-solve.

Test B: If A can do 80 pages in 18 minutes, then we know it can do 40 pages in 9 minutes. If it takes B 4 fewer minutes to do 40 pages, then B can do 40 pages in 5 minutes.
A's rate: 40/9 = 4 4/9 = About 4.4 pages per minute.
B's rate: 40/5 = 8 pages per minute
Combined rate = About 4.4+ 8 = 12.4 pages per minute. So in 6 minutes, the two could do 12.4 * 6 pages which is greater than 70. But they should be able to do 50 pages in this time, meaning they must be working slower. If they're working slower, it must have taken A more time to do 80 pages. Eliminate A and B.

Test D: If A can do 80 pages in 24 minutes, then we know it can do 40 pages in 12 minutes. If it takes B 4 fewer minutes to do 40 pages, then B can do 40 pages in 8 minutes.
A's rate: 40/12 = 10/3 pages per minute.
B's rate: 40/8 = 5 = 15/3 pages per minute
Combined rate = 10/3 + 15/3 = 25/3 pages per minute. In 6 minutes, the two would produce 6 * (25/3) = 2*25 = 50 pages. Bingo! D is our answer.
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by EconomistGMATTutor » Thu Dec 14, 2017 9:05 am
It takes printer A 4 more minutes more than printer B to print 40 pages. 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

OA is D
This question is confusing me.C an expert explain why D is the answer?
Hi Roland2rule,
Let's take a look at your question.

If printer B takes 'x' minutes to print 40 pages, then printer A will take 'x+4' minutes to print 40 pages.
$$\text{Rate of Printer A}=\frac{40}{x+4}$$
$$\text{Rate of Printer B}=\frac{40}{x}$$

$$\text{Combined Rate of Printer A and B}=\frac{40}{x+4}+\frac{40}{x} ... (i)$$

The question states, "Working together, the two printers can print 50 pages in 6 minutes".
$$\text{Combined Rate of Printer A and B}=\frac{50}{6} ... (ii)$$

Equating Eq(i) and (ii)
$$\frac{40}{x+4}+\frac{40}{x} =\frac{50}{6}$$
$$\frac{40x+40\left(x+4\right)}{x\left(x+4\right)}=\frac{50}{6}$$
$$\frac{40x+40x+160}{x\left(x+4\right)}=\frac{25}{3}$$
$$40x+40x+160=\frac{25x\left(x+4\right)}{3}$$
$$80x+160=\frac{25x\left(x+4\right)}{3}$$
$$3\left(80x+160\right)=25x\left(x+4\right)$$
$$240x+480=25x^2+100x$$
$$25x^2+100x-240x-480=0$$
$$25x^2-140x-480=0$$
$$5x^2-28x-96=0$$
$$5x^2-40x+12x-96=0$$
$$5x\left(x-8\right)+12\left(x-8\right)=0$$
$$\left(5x+12\right)\left(x-8\right)=0$$
$$Either\ \left(5x+12\right)=0,\ or\ \left(x-8\right)=0$$
$$Either\ x=-\frac{12}{5},\ or\ x=8$$

Since 'x' represents the number of minutes Printer B takes to print 40 pages,so it can not be negative. Therefore we will only consider x = 8.

To print 40 pages Printer A takes = x + 4 = 8 + 4 = 24

We are asked to find the number of minutes Printer A will take to print 80 pages .
= 2 * (Number of minutes Printer A will take to print 40 pages) = 2 * ( 12) = 24 minutes

Therefore, Option D is correct.

Hope it helps.
I am available if you'd like any follow up.
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by Jay@ManhattanReview » Fri Dec 15, 2017 12:06 am
Roland2rule wrote:It takes printer A 4 more minutes more than printer B to print 40 pages. 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

OA is D
This question is confusing me.C an expert explain why D is the answer?
Say printer B takes x minutes to print 40 pages, thus printer A would take (x + 4) minutes to print 40 pages.

We know that printer A and B together worked for 6 minutes and printed 50 pages, so let's compute the number of pages printer A and printer B would print in 6 minutes.

The number of pages printer A would print in 6 minutes = [40/(x + 4)]*6 = 240/(x + 4);
The number of pages printer B would print in 6 minutes = [40/x]*6 = 240/x

The number of pages printer A and B together would print in 6 minutes = 240/(x + 4) + 240/x

240/(x + 4) + 240/x = 50 pages (given)

=> 5x^2 - 28x - 96 = 0.

Instead of solving the unfamiliar linear equation, let's do the plug-in from the option values.

The option values are given for the time taken for printer A to print 80 pages. Since we assumed that printer A takes (x +4) minutes to print 40 pages, thus, it would take 2*(x + 4) minutes to print 80 pages. So the option values, in fact, are equal to 2(x + 4). Let's get the value of x from the options, and plug-in the equation 5x^2 - 28x - 96 = 0. If the RHS = the LHS, that option is the correct option.

A. 12: 2(x + 4) = 12 => x = 2. At x = 2, we have 5x^2 - 28x - 96 = 5*2^2 - 28*2 - 96 = 20 - 56 - 96 ≠ 0. Eliminated!
B. 18: 2(x + 4) = 18 => x = 5. At x = 5, we have 5x^2 - 28x - 96 = 5*5^2 - 28*5 - 96 = 125 - 140 - 96 ≠ 0. Eliminated!
C. 20: 2(x + 4) = 20 => x = 2. At x = 6, we have 5x^2 - 28x - 96 = 5*6^2 - 28*6 - 96 = 180 - 168 - 96 ≠ 0. Eliminated!
D. 24: 2(x + 4) = 24 => x = 2. At x = 8, we have 5x^2 - 28x - 96 = 5*8^2 - 28*8 - 96 = 320 - 224 - 96 = 320 - 320 = 0. Correct
E. 30: No need to check!

The correct answer: D

Hope this helps!

-Jay
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by Scott@TargetTestPrep » Mon Sep 23, 2019 5:10 pm
BTGmoderatorRO wrote:It takes printer A 4 more minutes more than printer B to print 40 pages. 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

OA is D
This question is confusing me.C an expert explain why D is the answer?

We can let x = the number of minutes printer B takes to print 40 pages. Thus, the rate of printer B is 40 / x and the rate of printer A is 40 / (x + 4). We can create the equation:

40 / x * 6 + 40 / (x+4) * 6 = 50

240 / x + 240 / (x+4) = 50

Multiplying by x(x+4) / 10, we have:

24(x+4) + 24x = 5x(x+4)

24x + 96 + 24x = 5x^2 + 20x

5x^2 - 28x - 96 = 0

(5x + 12)(x - 8) = 0

5x + 12 = 0 → x = -12/5 or x - 8 = 0 → x = 8

Since x can't be negative, x = 8, which means the rate of printer A is 40(x + 4) = 40/12 pages per minute. Therefore, it will take printer A 80/(40/12) = 80 * 12/40 = 2 * 12 = 24 minutes to print 80 pages.

Alternate Solution:

Let's first calculate the time it takes for the printers, individually and together, to print 400 pages (which is the LCM of 50 and 80).

Since working together, it takes them 6 minutes to print 50 pages; it will take them 6 x 8 = 48 minutes to print 400 pages.

To find the individual times, let t be the number of minutes it takes for printer B to print 400 pages. Since it takes printer A 4 minutes longer than printer B to print 40 pages, it will take printer A 4 x 10 = 40 minutes longer than printer B to print 400 pages.

Since printers A and B print 400 pages individually in (t + 40) and t minutes, respectively, and since they print the same number of pages in 48 minutes working together, we have:

1 / (t + 40) + 1 / t = 1/48

Multiplying each side by 48t(t + 40), we obtain:

48t + 48(t + 40) = t(t + 40)

48t + 48t + 48*40 = t^2 + 40t

96t + 48*40 = t^2 + 40t

t^2 - 56t - 48*40 = 0

Notice that 48*40 = 24*80 and 80 - 24 = 56; therefore, the equation can be factored as:

(t - 80)(t + 24) = 0

t = 80 or t = -24

Since t cannot be negative, it must be true that t = 80. Thus,, it takes printer A a total of (t + 40) = 80 + 40 = 120 minutes to print 400 pages, and so it will take 120/5 = 24 minutes for printer A to print 80 pages.

Answer: D

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