4.3: Improper fractions and mixed numbers

In the following, “A/B” represents the fraction “A over B.” For example, “4/3” refers to the fraction “4 over 3” or “four-thirds.”

Express an improper fraction as a mixed number

An improper fraction is a fraction that is greater than or equal to 1, i.e., its numerator is greater than or equal to its denominator. We can express an improper fraction as a mixed number (or mixed fraction) by thinking of the fraction in terms of dividing integers. For example,

• Express 13/5 (thirteen-fifths) as a mixed number.
• Think of 13/5 as 13 divided by 5.
• How many times does 5 go into 13?
  • 2 times with remainder 3 (since 2 x 5 = 10 and 13 – 10 = 3).
• Therefore, 13/5 = 2 + 3/5 = 2 3/5 (two and three-fifths).

Express a mixed number as an improper fraction

We can express a mixed number as an improper fraction by expressing the whole number part of the mixed number as a fraction and adding it to the fraction part using 2.5: Adding and subtracting fractions. For example,

• Express 2 3/5 (two and three-fifths) as an improper fraction.
• Think of 2 3/5 as 2/1 + 3/5.
• A common denominator for 1 and 5 is 5.
• So, 2/1 + 3/5 = (2×5)/(1×5) + 3/5 = 10/5 + 3/5 = 13/5
• Therefore, 2 3/5 = 13/5 (thirteen-fifths).

The video below works through some examples of expressing an improper fraction as a mixed number and vice versa.