The Book Fair Budget
A fourth-grade class earned $180 at a book fair. The teacher has a coupon: buy 4 books, get 1 free (the free book must be the lowest-priced of the 5). Books cost $8, $10, or $12 each. The class wants to buy at least 20 books and spend no more than $180. A student proposes buying 16 books at $10 each and using the coupon twice to get 2 free, totaling 18 books. Another student says that is not enough books and suggests buying only $8 books. The teacher asks the class to check both plans against the goals and find a plan that meets both constraints.
1. Evaluate the first student's plan: 16 books at $10 each with two coupon groups. Does it meet the goal of at least 20 books and a budget of $180? Show the arithmetic.
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Cost: 16 × $10 = $160. With two coupon groups of 5, the class gets 2 free books, so total books = 16 + 2 = 18. That is below 20, so the plan fails the book-count goal even though $160 ≤ $180. To reach 20 books, the class needs 18 paid books + 2 free = 20, costing 18 × $10 = $180. That exactly meets both goals, but leaves no margin. So the first plan is short by 2 books; buying 18 paid books at $10 works only if every book is $10.
2. The second student says to buy only $8 books. How many books can $180 buy with the coupon, and does this meet both goals? Explain any trade-off.
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At $8 each, $180 buys 22 paid books (22 × $8 = $176, with $4 left). Grouping in fives gives 4 complete groups of 5 (20 books) plus 2 extra, so 4 free books: total 22 + 4 = 26 books. This meets both goals easily. Trade-off: the class gets more books but they are all the cheapest title, which may not match student interests. The plan is valid mathematically; the judgement call is whether quantity or variety matters more.
3. Propose one plan that meets both goals and uses at least two different prices. Justify with numbers.
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Example: buy 10 books at $8 ($80), 6 at $10 ($60), and 2 at $12 ($24). Paid cost = $164. Arrange in coupon groups of 5; each group's free book is the lowest-priced in that group. With 18 paid books, form 3 groups of 5 (15 paid) plus 3 leftover, giving 3 free books, so total = 18 + 3 = 21 books. Cost stays $164 ≤ $180. This meets both goals and includes three price points. Students should verify the free book in each group is the cheapest, and that leftover books not in a group of 5 earn no free book.