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📘intermediate algebra review

Common Fractions

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

Common fractions, also known as simple fractions, are a way of representing a part of a whole using two numbers: a numerator and a denominator. The numerator indicates the number of parts being considered, while the denominator represents the total number of equal parts that the whole is divided into.

5 Must Know Facts For Your Next Test

  1. Common fractions can be used to represent and compare quantities, as well as perform mathematical operations such as addition, subtraction, multiplication, and division.
  2. The value of a common fraction is determined by the relationship between the numerator and denominator, with the numerator indicating the number of parts and the denominator indicating the total number of parts.
  3. Proper fractions are a type of common fraction where the numerator is less than the denominator, representing a value less than 1.
  4. Improper fractions are a type of common fraction where the numerator is greater than the denominator, representing a value greater than 1.
  5. Mixed numbers are a combination of a whole number and a proper fraction, allowing for the representation of values greater than 1.

Review Questions

  • Explain the relationship between the numerator and denominator in a common fraction and how it determines the value of the fraction.
    • In a common fraction, the numerator represents the number of parts being considered, while the denominator represents the total number of equal parts the whole is divided into. The value of the fraction is determined by the relationship between these two numbers. For example, in the fraction $\frac{3}{5}$, the numerator 3 indicates that 3 parts are being considered, and the denominator 5 indicates that the whole is divided into 5 equal parts. Therefore, the value of this fraction is 3 out of 5, or 0.6.
  • Describe the differences between proper fractions, improper fractions, and mixed numbers, and provide examples of each.
    • Proper fractions are common fractions where the numerator is less than the denominator, representing a value less than 1. For example, $\frac{2}{5}$ is a proper fraction. Improper fractions are common fractions where the numerator is greater than the denominator, representing a value greater than 1. For example, $\frac{7}{3}$ is an improper fraction. Mixed numbers are a combination of a whole number and a proper fraction, allowing for the representation of values greater than 1. For example, 2$\frac{1}{4}$ is a mixed number, which represents 2 and a quarter.
  • Explain how common fractions can be used to perform mathematical operations, such as addition, subtraction, multiplication, and division, and provide an example of each operation.
    • Common fractions can be used to perform various mathematical operations. For addition and subtraction, the fractions must have a common denominator, and the numerators are then added or subtracted accordingly. For example, to add $\frac{1}{4}$ and $\frac{3}{8}$, we first need to find a common denominator, which in this case is 8. The fractions then become $\frac{2}{8}$ and $\frac{3}{8}$, and the sum is $\frac{5}{8}$. For multiplication, the numerators are multiplied, and the denominators are multiplied. For example, $\frac{1}{3} \times \frac{2}{5} = \frac{2}{15}$. For division, the first fraction is multiplied by the reciprocal of the second fraction. For example, $\frac{1}{4} \div \frac{1}{2} = \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2}$.