Equation solving math puzzle

Question

An office used 3 types of envelopes: small, medium, large. They used 80 small envelopes. The no. of medium envelopes used is 5 times the no. of small envelopes plus 5/20 of the total number of envelopes. The no. of large envelopes was equal to the no. of small envelopes plus 5/20 of the no. of medium envelope. What is the number of total envelopes?

Answer 1

The hard part of this problem is deciphering this word problem. I will help you with that and let you solve the system at the end.

 

Let S, M, and L be the number of small, medium, and large envelopes, respectively. Then we know that S = 80 <-- keep this in mind for the rest of the set up.

 

Now we will break down each statement and translate it:

   The no. of medium envelopes used (M) is (=) 5 times (5*) the no. of small envelopes (80) plus (+) 5/20 of the total number of envelopes [5/20 * (S + M + L)]

   M = 5*80 + 5/20*(S + M + L)

   M = 5*80 + 1/4*(80 + M + L)

 

Simplify this expression to get:

   M = 5*80 + 1/4*(80 + M + L)

   M = 400 + 20 + 1/4*M + 1/4*L

   M = 420 + 1/4*M + 1/4*L

   -420 = -3/4*M + 1/4*L <-- This is the first equation

 

Now for the next sentence:

   The no. of large envelopes (L) was equal to (=) the no. of small envelopes (80) plus (+) 5/20 of the no. of medium envelopes (5/20*M)

   L = 80 + 5/20*M <-- This is the second equation

 

To finish the problem you want to solve the first and second equations by substituting the second one into the first. Can you handle it from there? I hope this helps.   

Answer 2

Avi, let's look at what we have.

 

We have 80 small, at least 5(80) = 400 medium and at least 80 large, plus 5/20 + 5/20 = 10/20, or 1/2 of the total, for a total of 560 + 1/2 of the total number of envelopes.

 

Well, if we are adding half the total, that means 560 is half of the total already, so there are 560 + 560 = 1120 envelopes.

 

It doesn't ask, but 80 small, 400 + 280 = 680 medium, and 80 + 280 = 360 large.