A shcol copier can print 5 copies of a test per min and 8 copies of a quiz a min. If i make 167 copies in 25 min how many copies did i print?

Question

I need to show my work.

Answer 1

Hi Marie,

 

The copier printed test pages at 5 pages/minute for t minutes. Then, it printed quiz pages at 8 pages per minute for (25 - t) minutes. Altogether, 167 pages were printed.

 

(5 test pages/minute)(t minutes) + (8 quiz pages / minute)(25 - t  min) = 167 

 

Notice that minutes cancel out and each side of the equation just counts pages.

 

Neglecting the unit labels, we write,

 

5t + 8(25 - t) = 167

 

or

 

5t + 200 - 8t = 167

 

Then,

 

-3t = -33

 

Or

 

3t = 33

 

Then, t = 11 minutes

 

So, test pages were printed for 11 minutes at a rate of 5 pages per minute  for a total of (11)(5) = 55 test pages.

 

Quiz pages were printed out for (25 - 11) = 14 minutes at 8 pages per minute for (14)(8) = 112 quiz pages

 

To check this answer, altogether we have printed 55 + 112 = 167 pages.

Answer 2

In 1 min the copier can print 5 copies of a test

In 25 min, same copier can print 125 copies of a test, by rule of proportion 

But I have made 167 copies in 25 min which is (167-125)=42 more copies than the number of copies of test the the copier can print in 25 min.

Hence, I must have printed some copies of a quiz.

In 1 min the copier can print 8 copies of the quiz

In 25 min, same copier can print 200 copies of the quiz, by rule of proportion

But I have made 167 copies in 25 min which is (200-167)=33 less copies than the number copies of quiz the the copier can print in 25 min.
Hence, I must have printed some copies of the quiz and some copies of the test.

In 1 min the copier can print 5 copies of a test, i.e odd number of copies and 8 copies of a quiz, i.e., even number of copies. The total number of copies being 167, an odd number, clearly some odd minutes of 25 minutes I have made copies of test and rest of the time copies of quiz.

In 1 minute the copier can print (8-5) = 3 more copies of quiz than the number copies of test
In other words, the copier can print 3 more copies of quiz than the number copies of test in 1 minute
Hence, by rule of proportion, the copier can print 42  more copies of quiz than the number of copies of test in 42/3 = 14 minutes 

In 14 minutes the copier can print, 112 copies of a quiz, by rule of proportion
And in remaining (25-14)=11 minutes the copier can print, 55 copies of the test, by rule of proportion
Thus in (14+11) = 25 minutes the copier can print, 112 copies of the quiz and  55 copies of a test or (112+55)=167 copies in total

Again,

In 1 minute the copier can print (8-5) = 3 less copies of test  than the number copies of quiz 

In other words,  the copier can print 3 less copies of test than the number copies of quiz in 1 minute

Hence,  by rule of proportion,  the copier can print 33 less copies of test than the number copies of quiz in  33/3 = 11 minutes 

In 11 minutes the copier can print, 55 copies of a test, by rule of proportion

And in remaining (25-11)=14 minutes the copier can print, 112 copies of the quiz, by rule of proportion 

Thus in (11+14) = 25 minutes the copier can print, 55 copies of a test and 112 copies of the quiz or (55+112)=167 copies in total

Therefore, I printed 55 copies of a test and 112 copies of the quiz.