# 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: $12 X + 5 Y = 60$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 $5 Y = 60 - 12 X$or (divide by $5$) $Y = 12 - 12 \cdot \frac{X}{5}$, 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 $Y = 0$.