A school copier can print 5 copies of a test per minute and 8 copies of a quiz per minute, If I make 167 copies in 25 minutes, how many of each did I copy?

Question

I would like a descriptive breakdown either via the substitution method or elimination method.

Answer 1

Hello April, let's begin by setting up the equations you will need: Let t and q be the number of copies of tests and quizzes respectively.  Then one obvious equation is:

(1) : t+q=167

 

Our other equation should use the fact about 5 copies of a test per minute vs 8 copies of a quiz per minute:

(2) : t/5+q/8 =25

 

These are the two equations to use.  To solve them via the elimination method you must manipulate the equations so that adding them will eliminate a variable.  The easy to do this is to multiply (2) by 5 and 8 first to get rid of fractions, so in effect you'll be multiplying by 40 giving you:

(3)  : 8t +5q = 1000

Now multiply (1) by 5 and get:

(4) : 5t+5q = 167*5   (You'll see why I dont bother multiplying this out even though it is not difficult)

.  Now subtract (4) from (3) as follows:

 

  8t +5q= 1000

- 5t+5q = 167*5

3t = 1000 - 167*5

3t = 5(200) - 5(167) = 5(33)

t = 5*33/3 = 5*11 = 55

 

From this you can easily work out what q is.

 

Now for the subsitution method.  We go back to (1) and (2) which are: 

 

(1) : t+q=167

(2) : t/5+q/8 =25

 

Solve for t in (1):
t=167-q

Substitute into (2):

(167 - q)/5 +q/8 = 25

Again multiply by 40:

8(167-q) +5q = 1000

8*167-8q +5q=1000

-3q = 1000-8*167 = 125*8 -167*8  = 8(125-167) = 8(-42) = 8(-42)/(-3) = 8*14

q = 112

 

Again from here you can substitute back into (1) or (2) to get the value of t.  (Check both methods to make sure you get the same values.)

 

You can see that I avoid doing needless multiplication in favor of knowing when factoring numbers is going to be easier.  My way is a bit slower since there are a few steps where I am just manipulating numbers and not really doing anything to the equation but it makes the numbers easier to think about when you dont have a calculator handy and don't want to work out products and quotients until you have to.

 

Hope this helped.