Problem 9 - Prealgebra

Question

Fifty-four of the 90 books on a shelf are history books. What percent of the books are history books?

Step-by-Step Solution

Answer

60% of the books are history books.

Step 1: Understand the Problem

We are given that there are 90 books in total on a shelf, and 54 of these are history books. We need to determine what percent of these books are history books.

Step 2: Define Percentage Formula

Remember that percentage is calculated by dividing the part by the whole and then multiplying by 100. The formula for percentage is: \[ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \% \] In this problem, the part is the number of history books (54), and the whole is the total number of books (90).

Step 3: Substitute Values into Formula

Substitute the given values into the percentage formula: \[ \text{Percentage} = \left( \frac{54}{90} \right) \times 100 \% \]

Step 4: Perform the Calculation

Calculate the fraction first: \[ \frac{54}{90} = 0.6 \] Next, multiply by 100 to convert the decimal to a percentage: \[ 0.6 \times 100 = 60 \% \]

Step 5: Interpret the Result

The calculation shows that 60% of the books on the shelf are history books. Therefore, 60% of the books are history books.

Key Concepts

Percentage Calculation

Understanding how to calculate percentages is a useful skill, particularly when dealing with data analysis or when you want to express one quantity as a part of a whole. In essence, a percentage represents a fraction of 100. This provides a way to standardize values, making it easy to compare different quantities.

To compute a percentage, you use a simple mathematical formula:

• Identify the "part" – the quantity you're interested in. In our example, it's the 54 history books.
• Determine the "whole" – the total , such as the total books on the shelf, which is 90.
• Use the formula: \( ext{Percentage} = rac{ ext{Part}}{ ext{Whole}} imes 100 \)

Substituting the given values into this formula provides the percentage of history books: \[ ext{Percentage} = rac{54}{90} imes 100 = 60\% \]
This means that 60 out of every 100 books on the shelf are history books.

Fractions

Fractions are mathematical expressions representing parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). In our problem, the fraction \( \frac{54}{90} \) represents the part of the shelf filled with history books.

The numerator (54) shows the number of favorable parts, whereas the denominator (90) represents the total possible parts.

Fractions can often be simplified by finding a common factor. Here, both 54 and 90 are divisible by 18, allowing you to simplify the fraction to \( \frac{3}{5} \).

• Simplifying makes it easier to interpret or compare with other fractions.
• Understanding fractions helps in converting them to decimals or percentages, expanding their usability across different math problems.

By recognizing how fractions translate into percentages, you can better manipulate and understand data.

Basic Arithmetic Operations

Basic arithmetic operations are fundamental in solving mathematical problems. They include addition, subtraction, multiplication, and division. In our example, we utilize division and multiplication.

First, you divide to find the fraction, which is \( \frac{54}{90} = 0.6 \). Division helps you transform parts to whole numbers or decimals with ease. Multiplication follows to help convert that decimal to a percentage. Multiplying by 100 shifts the decimal so that it's expressed as parts per hundred:

• This operation highlights the transition from a simple ratio to a readable format like percentages.
• It is crucial in scenarios where comparison or data interpretation is needed. Using these basic operations strengthens your mathematical foundation and problem-solving skills.

Learning these arithmetic operations ensures you can tackle more advanced topics with confidence.