In a library, the ratio of science books to literature books was initially 4 : 5. After 60… — JAMB UTME

In a library, the ratio of science books to literature books was initially $4 : 5$ . After $60$ new science books were added to the collection and $20\%$ of the literature books were donated to a local school, the ratio of science books to literature books became $3 : 2$ . What was the total number of books (science and literature combined) originally in the library?

A
$120$
B
$150$
$270$ Cevap
D
$300$

Cevap

The original total number of books in the library was

270

Let the original number of science books be

4x

and literature books be

5x

. After adding

60

science books, there are

4x + 60

science books. Donating

20\%

of literature books leaves

80\%

5x

, which equals

4x

literature books. Setting up the ratio

\frac{4x + 60}{4x} = \frac{3}{2}

yields

8x + 120 = 12x

, so

4x = 120

and

x = 30

. The original total number of books was

4x + 5x = 9x = 9(30) = 270

Adım Adım Çözüm

Define initial quantities using a ratio multiplier

x

Initial science books =

4x

, initial literature books =

5x

, and total original books =

9x

The initial ratio given is

4 : 5

Calculate the updated quantities after books are added and donated.

New science books =

4x + 60

; new literature books =

5x \times (1 - 0.20) = 4x

60

science books were added and

20\%

of literature books were donated.

Set up the proportion for the new ratio of

3 : 2

\frac{4x + 60}{4x} = \frac{3}{2}

The new ratio of science to literature books is stated as

3 : 2

Solve the algebraic proportion for

x

2(4x + 60) = 3(4x) \implies 8x + 120 = 12x \implies 4x = 120 \implies x = 30.

Cross-multiplication simplifies the proportional equation to find the value of

x

Compute the original total number of books.

9x = 9 \times 30 = 270.

The original total collection consists of

9x

books.

Anahtar Kavram

Solving multi-step ratio changes and proportions with percentage alterations

Tahmini Süre:1m 30s

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