In a community library, \frac{4}{15} of the total book collection consists of fiction book… — GRE General Test

In a community library, $\frac{4}{15}$ of the total book collection consists of fiction books, $\frac{2}{5}$ consists of non-fiction books, and the remaining 240 books are children's books. If $\frac{3}{8}$ of the fiction books are hardcovers and the rest are softcovers, how many softcover fiction books are in the library?

Answer: 120 books

Answer

The total number of softcover fiction books in the library is 120.

To find the number of softcover fiction books, first calculate the total number of books in the library. The combined fraction of fiction and non-fiction books is

\frac{4}{15} + \frac{2}{5} = \frac{4}{15} + \frac{6}{15} = \frac{10}{15} = \frac{2}{3}

. Therefore, children's books represent the remaining

1 - \frac{2}{3} = \frac{1}{3}

of the collection. Since

\frac{1}{3}

of the total equals 240 books, the total collection is

240 \times 3 = 720

books. Next, compute the total number of fiction books:

\frac{4}{15} \times 720 = 192

. Finally, since

\frac{3}{8}

of fiction books are hardcovers, the remaining fraction of softcovers is

1 - \frac{3}{8} = \frac{5}{8}

. Multiplying

\frac{5}{8} \times 192

gives 120 softcover fiction books.

Step-by-Step Solution

Find the combined fraction of fiction and non-fiction books, then determine the fraction of children's books.

Combined fraction =

\frac{4}{15} + \frac{6}{15} = \frac{10}{15} = \frac{2}{3}

. Children's books fraction =

1 - \frac{2}{3} = \frac{1}{3}

The sum of all non-overlapping fractions comprising the entire collection must equal 1.

Calculate the total number of books in the library.

Total collection =

240 \div \frac{1}{3} = 720

books.

The 240 children's books represent exactly

\frac{1}{3}

of the whole collection.

Determine the total count of fiction books.

Fiction books =

\frac{4}{15} \times 720 = 192

books.

Fiction books make up

\frac{4}{15}

of the total 720 books.

Calculate the number of softcover fiction books.

Softcover fiction books =

\left(1 - \frac{3}{8}\right) \times 192 = \frac{5}{8} \times 192 = 120

books.

\frac{3}{8}

of the fiction subset are hardcovers, the remaining

\frac{5}{8}

of that subset are softcovers.

Key Concept

Solving multi-step rational number word problems by combining fractions and determining fractional parts of a subset.

Estimated Time:1m 30s

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