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Chapter 1: Problem 52

A closet containing 50 shoes has 12 that are black. What percent of the shoes are black?

Short Answer

Expert verified
24%

Step by step solution

01

Identify the Total Number of Shoes

The problem states that there are 50 shoes in total.
02

Identify the Number of Black Shoes

The problem also states that 12 of the shoes are black.
03

Calculate the Fraction of Black Shoes

Use the information from Step 1 and Step 2 to set up the fraction of black shoes: \[ \frac{12}{50} \]
04

Convert the Fraction to a Percentage

To find the percentage, multiply the fraction by 100: \[ \left( \frac{12}{50} \right) \times 100 = 24\% \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction to Percentage Conversion
Understanding how to convert a fraction to a percentage is very useful in various math problems. Start by writing down the fraction you need to convert. In this case, the fraction of black shoes is \( \frac{12}{50} \). To convert a fraction to a percentage, perform the following steps:
1. Divide the numerator by the denominator to get a decimal, which in this case is \( \frac{12}{50} = 0.24 \).
2. Multiply the decimal by 100 to get the percentage: \( 0.24 \times 100 = 24\).
This means that 24% of the shoes in the closet are black.
Remember, multiplying by 100 is equivalent to moving the decimal point two places to the right.
Basic Algebra
Basic algebra involves using variables and numbers to solve math problems. In this exercise, we used basic algebra by setting up the problem as a fraction and converting it to a percentage. Here are a few key concepts:
  • Numerators and denominators: In the fraction \( \frac{12}{50} \), 12 is the numerator (top number) and 50 is the denominator (bottom number).
  • Operations: Addition, subtraction, multiplication, and division are basic operations.
  • Proportions: We often use proportions to solve for unknowns, like finding the percentage of black shoes.
By understanding these foundational concepts, you can apply similar steps to different algebra problems.
Proportion
A proportion is an equation that states two ratios are equal. In the problem about the shoes, we are essentially dealing with proportions. Here鈥檚 how it works:
1. Recognize the ratio of black shoes to total shoes: \( \frac{12}{50} \)
2. Convert this ratio to a proportion by setting it equal to a percentage (where percentages are out of 100): \( \frac{12}{50} = \frac{x}{100} \).
3. Solve for \( x \) by multiplying both sides by 100 to clear the denominator: \( 12 \times 2 = 24 \)
Therefore, \( x = 24 \), meaning 24% of the shoes are black.
Using proportions helps simplify complex problems and can be applied to various scenarios, like mixing ingredients or calculating distances.

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