A community library purchases two types of books: hardcover books for \24 each and paperba… — GRE General Test

A community library purchases two types of books: hardcover books for $\$24$ each and paperback books for $\$15$ each. The library spends a total of $\$360$ on these books and purchases at least one book of each type. Which of the following could be the total number of books purchased? Select all such numbers.

A
15
18Answer
C
20
21Answer
E
24

Answer

18 and 21

The linear modeling equation

24x + 15y = 360

simplifies to

8x + 5y = 120

. Since

x

and

y

must be positive integers,

x

must be a multiple of 5. The only valid solutions satisfying

x \ge 1

and

y \ge 1

are

(x=5, y=16)

and

(x=10, y=8)

, which yield total book counts of 21 and 18, respectively.

Step-by-Step Solution

Set up the linear equation from the given word problem context.

24x + 15y = 360

, where

x \ge 1

is the number of hardcover books and

y \ge 1

is the number of paperback books, with

x, y \in \mathbb{Z}^+

Total expenditure is the sum of cost per hardcover times number of hardcovers plus cost per paperback times number of paperbacks.

Simplify the equation by dividing both sides by the greatest common divisor, 3.

8x + 5y = 120

Simplifying coefficients reduces arithmetic complexity and isolates integer conditions.

Express

y

in terms of

x

to identify valid integer pairs

(x, y)

y = \frac{120 - 8x}{5} = 24 - \frac{8x}{5}

For

y

to be an integer,

8x

must be divisible by 5, meaning

x

must be a positive multiple of 5.

Test valid positive integer values for

x

such that

y \ge 1

x = 5

y = 24 - 8 = 16

, giving total books

x + y = 21

. If

x = 10

y = 24 - 16 = 8

, giving total books

x + y = 18

. If

x \ge 15

y \le 0

, which is invalid.

These are the only integer solutions satisfying

x \ge 1

and

y \ge 1

Key Concept

Linear Diophantine Equations in Word Problems

Estimated Time:1m 30s

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