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Difficulty: MediumSystems of Linear Inequalities in Two Variables

A public library is purchasing new books for its collection. The library plans to buy paperback books for 6eachandhardcoverbooksfor6 each and hardcover books for 20 each. The library has a budget of at most $1,200 for this purchase, wants to buy at most 120 total books, and must purchase at least 45 hardcover books. What is the maximum number of paperback books the library can purchase?

Answer: 50 books

Answer

The maximum number of paperback books the library can purchase is 50.
The correct maximum number of paperback books is 50. By expressing the budget constraint as p≤200−103hp \le 200 - \frac{10}{3}h, we see that the number of paperback books is maximized when the number of hardcover books, hh, is minimized. Since the library must buy at least 45 hardcover books, we substitute h=45h = 45 into the inequality to get p≤50p \le 50. This combination also satisfies the total book constraint because 50+45=9550 + 45 = 95, which is less than or equal to 120.

Step-by-Step Solution

1
Set up the system of inequalities representing the library's constraints.
Let pp be the number of paperbacks and hh be the number of hardcovers. The constraints are 6p+20h≤12006p + 20h \le 1200, p+h≤120p + h \le 120, and h≥45h \ge 45.
This translates the word problem into mathematical inequalities.
2
Isolate the variable pp in the budget inequality to express it in terms of hh.
p≤200−103hp \le 200 - \frac{10}{3}h
This shows how the number of paperbacks depends on the number of hardcovers purchased.
3
Maximize pp by using the minimum possible value of hh.
p≤200−103(45)=50p \le 200 - \frac{10}{3}(45) = 50
To maximize pp, we must minimize hh because buying more hardcovers decreases the remaining budget for paperbacks. The minimum value for hh is 45.
4
Verify if the solution satisfies the remaining total books constraint.
50+45=95≤12050 + 45 = 95 \le 120, which is true.
This ensures the solution is feasible and does not violate any other constraints.

Key Concept

Systems of Linear Inequalities in Two Variables
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Systems of Linear Inequalities in Two Variables

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