Question 1

Question

Santiago is working at his mom’s day care this summer. His mom has asked him to check out the
library book sale to look for books they might be able to use in the day care. When he gets to the
book sale, he sees that board books are $1.50 and soft cover books are $0.50. He picks out a
variety of books and spends $20.
A. Write a two-variable equation to represent this situation. Be sure to define the variables
you use in your equation.
B. Is it possible that Santiago bought 10 board books and five soft cover books? Use your
equation from part A to check. Explain your answer. in 1 sentence
C. What are the constraints of the situation? Describe them.
D. Make a list of all viable solutions to the problem.
E. Suppose Santiago bought 34 books. Can you figure out how many of each type of book he
bought? Explain. In 1 sentence

Answer 1

A. Let $b$ be the number of board books and $s$ be the number of soft cover books. The two-variable equation representing the situation is:
$$1.50b+0.50s=20$$

B. No, it is not possible that Santiago bought 10 board books and five soft cover books because substituting these values gives $1.50(10)+0.50(5)=15+2.5=17.5$, which is less than $20.

C. The constraints of the situation are that the number of board books ($b$) and the number of soft cover books ($s$) must be non-negative integers (i.e., $b\ge 0$ and $s\ge 0$) and their combined total price must equal $20.

D. To find all viable solutions, we can express $s$ in terms of $b$ using the original equation:
$$s=40-3b$$
The solutions must yield non-negative integers for both $b$ and $s$. Therefore, possible pairs $(b,s)$ that satisfy the equation with $b$ being from 0 to 13 (as 14 board books alone would exceed $20) would be:

• $(0,40)$
• $(1,37)$
• $(2,34)$
• $(3,31)$
• $(4,28)$
• $(5,25)$
• $(6,22)$
• $(7,19)$
• $(8,16)$
• $(9,13)$
• $(10,10)$
• $(11,7)$
• $(12,4)$
• $(13,1)$

E. Yes, if Santiago bought 34 books, then he could be described using the system of equations:
$$b+s=34$$ and $$1.50b+0.50s=20$$. By solving these equations simultaneously, we find $b=8$ (board books) and $s=26$ (soft cover books).