Suppose you intend to spend $60 buying books. Hardback books cost $12 and paperback cost $5. How many books of each type can you buy?

1 Answer
Sep 8, 2015

Answer:

We have two solutions:
A) All money ($60) are spent on 12 paperbacks at $5 each
B) All money ($60) are spent on 5 hardbacks at $12 each

Explanation:

Let's approach this as a problem with two unknown:
#X# - number of books at $12 per book and
#Y# - number of books at $5 per book.

There is only one equation that these two variables satisfy:
#12X+5Y = 60#

Generally speaking, one equation is insufficient to find a solution for two variables among all real numbers. However, we should not look for real numbers as solutions, only for positive integer ones.

Since #Y# is integer and our equation can be presented as
#5Y=60-12X# or (divide by #5#)
#Y=12-12*X/5#,
the number of books at $12 (that is, #X#) must be a multiple of #5#.
Therefore, #X# can be either #0# (and then all money are spent on 12 books at $5 each) or it can be #5# (and then all money are spent on these 5 books at $12 each). No other solutions exist.

So, we have two solutions:
#X=0# and #Y=12#
#X=5# and #Y=0#.