Question

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?

Answer 1

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`.