Mixed Numbers and Improper Fractions: What’s the Difference?

An improper fraction being turned into a mixed number.
An improper fraction being turned into a mixed number.

Fractions are used to represent parts of a whole, but once a fraction becomes larger than 1, there are two common ways to write it: as an improper fraction or as a mixed number. These two forms often represent exactly the same quantity; the difference is simply how the number is written.

Improper fractions

An improper fraction is a single fraction where the numerator is greater than or equal to the denominator. This means the fraction is at least 1 whole (and possibly more).

Examples:

  • 7/4
  • 12/5
  • 9/9

In 7/4, there are 7 parts, and each whole is made from 4 parts. Since 7 is larger than 4, the value is greater than 1.

Mixed numbers

A mixed number is written as a whole number followed by a proper fraction. The proper fraction shows the leftover part after counting whole units.

Examples:

  • 1 3/4
  • 2 2/5
  • 3 1/6

In 1 3/4, the “1” shows one whole, and “3/4” shows three extra quarters.

Same value, different form

Improper fractions and mixed numbers can represent the same quantity. For example, 7/4 and 1 3/4 are equal.

To see why, note that:

  • 4/4 makes 1 whole
  • 7/4 = 4/4 + 3/4 = 1 + 3/4 = 1 3/4

This shows that the value does not change when switching form; only the notation changes.

Why the word “improper” is misleading

The term “improper fraction” does not mean the fraction is incorrect. It is simply the standard name for fractions whose numerator is at least as large as the denominator. In fact, improper fractions are often preferred in calculations because they are written as one fraction, which makes arithmetic and algebra simpler.

How to convert between the two

Improper fraction → mixed number

Divide the numerator by the denominator. The quotient is the whole-number part, and the remainder becomes the new numerator.

Example: 19/6

  • 19 ÷ 6 = 3 remainder 1
  • So 19/6 = 3 1/6

Example: 7/4

  • 7 ÷ 4 = 1 remainder 3
  • So 7/4 = 1 3/4

Mixed number → improper fraction

Multiply the whole number by the denominator, then add the numerator. Keep the same denominator.

Rule:

(whole number × denominator + numerator) / denominator

Example: 3 1/6

  • (3 × 6 + 1) / 6 = (18 + 1) / 6 = 19/6

Example: 1 3/4

  • (1 × 4 + 3) / 4 = 7/4

When each form is useful

Mixed numbers are useful in everyday contexts such as measurements, because the whole number makes the size easy to read quickly. For example, 2 1/3 litres is immediately understood as “a bit more than 2 litres”.

Improper fractions are useful for arithmetic and algebra, because calculations are usually easier with a single fraction. For example, adding 7/4 + 5/4 is straightforward because the denominators already match.

Common mistakes to avoid

  • Forgetting the remainder when converting an improper fraction to a mixed number. For example, 19/6 is 3 1/6, not 3 3/6.
  • Forgetting to multiply the whole number by the denominator when converting a mixed number to an improper fraction. For example, 2 3/5 is 13/5, not (2+3)/5.

Once these two forms are understood, converting between them becomes routine, and choosing the most useful form for a task becomes a simple decision: mixed numbers for readability, improper fractions for calculation.

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