Problem Solving

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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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