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Difficulty: Very hardRatio, Proportion, and Rate

In a library, the ratio of science books to literature books was initially 4:54 : 5. After 6060 new science books were added to the collection and 20%20\% of the literature books were donated to a local school, the ratio of science books to literature books became 3:23 : 2. What was the total number of books (science and literature combined) originally in the library?

  1. A
    120120
  2. B
    150150
  3. 270270Answer
  4. D
    300300

Answer

The original total number of books in the library was 270270.
Let the original number of science books be 4x4x and literature books be 5x5x. After adding 6060 science books, there are 4x+604x + 60 science books. Donating 20%20\% of literature books leaves 80%80\% of 5x5x, which equals 4x4x literature books. Setting up the ratio 4x+604x=32\frac{4x + 60}{4x} = \frac{3}{2} yields 8x+120=12x8x + 120 = 12x, so 4x=1204x = 120 and x=30x = 30. The original total number of books was 4x+5x=9x=9(30)=2704x + 5x = 9x = 9(30) = 270.

Step-by-Step Solution

1
Define initial quantities using a ratio multiplier xx.
Initial science books = 4x4x, initial literature books = 5x5x, and total original books = 9x9x.
The initial ratio given is 4:54 : 5.
2
Calculate the updated quantities after books are added and donated.
New science books = 4x+604x + 60; new literature books = 5x×(1−0.20)=4x5x \times (1 - 0.20) = 4x.
6060 science books were added and 20%20\% of literature books were donated.
3
Set up the proportion for the new ratio of 3:23 : 2.
4x+604x=32\frac{4x + 60}{4x} = \frac{3}{2}
The new ratio of science to literature books is stated as 3:23 : 2.
4
Solve the algebraic proportion for xx.
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 xx.
5
Compute the original total number of books.
9x = 9 \times 30 = 270.
The original total collection consists of 9x9x books.

Key Concept

Solving multi-step ratio changes and proportions with percentage alterations
Estimated Time:1m 30s
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Topics

Ratio, Proportion, and Rate

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