A public library is organizing its catalog of physical books. Currently, \frac{3}{8} of th… — ACT

A public library is organizing its catalog of physical books. Currently, $\frac{3}{8}$ of the books in the collection are categorized as fiction. Of the remaining books, $40\%$ are non-fiction, while the rest are divided between children's literature and reference volumes in a ratio of $3:2$ , respectively. If there are exactly $450$ reference volumes in the library, what is the total number of physical books in the collection?

A
$1,350$
B
$1,800$
C
$1,875$
$3,000$ Cevap
E
$5,000$

Cevap

The total number of physical books in the library collection is

3,000

To find the total number of books, first calculate the remaining fraction after fiction books are removed:

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

. Next, account for non-fiction taking up

40\%

of this remainder, leaving

60\%

(or

\frac{3}{5}

) of

\frac{5}{8}

for children's and reference books, which equals

\frac{3}{8}

of the entire collection. Using the

3:2

ratio, reference books represent

\frac{2}{5}

of this

\frac{3}{8}

portion, which is

\frac{3}{20}

of the total collection. Setting

\frac{3}{20}

of the total equal to

450

gives a total collection of

3,000

books.

Adım Adım Çözüm

Determine the remaining fraction of books after setting aside fiction books.

Since

\frac{3}{8}

of the books are fiction, the remaining fraction is

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

The total collection equals

1

(or

100\%

). Subtracting the fiction fraction leaves the remaining portion.

Calculate the fraction of books that are either children's literature or reference volumes.

Since

40\%

of the remaining books are non-fiction, the remaining

60\%

(

0.60

\frac{3}{5}

) of the

\frac{5}{8}

portion are children's or reference books:

\frac{3}{5} \times \frac{5}{8} = \frac{3}{8}

Finding

60\%

of the remaining

\frac{5}{8}

identifies the portion shared between children's and reference books.

Determine the fraction of total books represented by reference volumes using the given ratio.

With a children's to reference ratio of

3:2

, reference books represent

\frac{2}{3+2} = \frac{2}{5}

of this subgroup. Thus, reference books account for

\frac{2}{5} \times \frac{3}{8} = \frac{6}{40} = \frac{3}{20}

of the total collection.

A ratio of

3:2

divides the subgroup into

5

equal parts, of which reference books make up

2

parts.

Solve for the total collection size using the known number of reference volumes.

Set up the equation

\frac{3}{20} \times T = 450

, where

T

is the total number of books. Solving yields

T = 450 \times \frac{20}{3} = 150 \times 20 = 3,000

Dividing the given quantity by its corresponding fraction gives the total population size.

Anahtar Kavram

Fractions, Decimals, and Percentages

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