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Problem 3

# The librarian is rearranging the books on the shelves. Now, $$0.4$$ of the shelves are adult fiction, $$38 \%$$ of the shelves hold nonfiction, and $$\frac{9}{20}$$ of the shelves hold children's books. Which type of book uses the most shelf space?

Expert verified
Children's books use the most shelf space, as they occupy $$\frac{45}{100}$$ of the shelves, which is larger than the fractions representing adult fiction ($$\frac{40}{100}$$) and nonfiction ($$\frac{38}{100}$$).
See the step by step solution

## Step 1: Convert all values into fractions

First, we need to make sure everything is in the same format. The three values given are in decimal, percentage and fraction. Let's convert the decimals and percentages into fractions to make it easier to compare them. For adult fiction, we have 0.4, which is $$0.4 = \frac{2}{5}$$. For nonfiction, we have 38%, which, when written as a fraction, is $$\frac{38}{100}$$, which can be simplified to $$\frac{19}{50}$$. For children's books, the fraction is already given, $$\frac{9}{20}$$.

## Step 2: Find a common denominator for all the fractions

In order to compare the fractions, we need a common denominator. We can find the least common multiple (LCM) of the denominators 5, 50, and 20 to find a common denominator. The prime factors of 5 are 5. The prime factors of 50 are 2, 5, and 5. The prime factors of 20 are 2, 2, and 5. The LCM is found by multiplying the highest power of each prime factor: $$2^2 * 5^2 = 4 * 25 = 100$$. So, the common denominator is 100.

## Step 3: Convert the fractions to have the common denominator

Now, we need to convert each fraction to an equivalent one with a denominator of 100. For adult fiction: $$\frac{2}{5}\cdot \frac{20}{20} = \frac{40}{100}$$ For nonfiction: $$\frac{19}{50}\cdot \frac{2}{2} = \frac{38}{100}$$ For children's books: $$\frac{9}{20}\cdot \frac{5}{5} = \frac{45}{100}$$

## Step 4: Compare the fractions and find the largest one

Now that all the fractions have the same denominator, we can compare their numerators to see which one is the largest. Adult fiction: $$\frac{40}{100}$$ Nonfiction: $$\frac{38}{100}$$ Children's books: $$\frac{45}{100}$$ The largest fraction is $$\frac{45}{100}$$, which represents the shelf space used by children's books. Therefore, the type of book that uses the most shelf space is children's books.

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