Mixed numbers and improper fractions are two ways to describe quantities greater than or equal to one whole. A mixed number uses a whole number and a proper fraction, such as 2 1/3. An improper fraction uses one fraction whose numerator is greater than or equal to its denominator, such as 7/3.
Knowing how to move between these forms makes fraction operations easier and helps you choose the clearest form for an answer.
To convert a mixed number to an improper fraction, count how many fractional pieces are in the whole parts, then add the leftover pieces. To convert an improper fraction to a mixed number, divide the numerator by the denominator and use the remainder as the new numerator. Addition and subtraction often work best by converting to improper fractions or by combining whole parts and fraction parts carefully.
Multiplication of mixed numbers is usually simplest after converting every mixed number to an improper fraction.
Key Facts
- Mixed number form: a b/c means a wholes plus b/c of another whole.
- Improper fraction form: numerator is greater than or equal to denominator, such as 9/4 or 4/4.
- Convert mixed to improper: a b/c = (a × c + b)/c.
- Convert improper to mixed: n/d = quotient remainder/d after dividing n by d.
- Add or subtract fractions with like denominators: a/c + b/c = (a + b)/c and a/c - b/c = (a - b)/c.
- Multiply mixed numbers by converting first: a b/c × d e/f = ((a × c + b)/c) × ((d × f + e)/f).
Vocabulary
- Mixed number
- A number written as a whole number together with a proper fraction, such as 3 2/5.
- Improper fraction
- A fraction whose numerator is greater than or equal to its denominator, such as 11/6.
- Numerator
- The top number in a fraction that tells how many equal parts are being counted.
- Denominator
- The bottom number in a fraction that tells how many equal parts make one whole.
- Remainder
- The amount left over after division that becomes the numerator of the fractional part in a mixed number.
Common Mistakes to Avoid
- Adding the whole number only to the numerator when converting a mixed number is wrong because 3 1/4 is not 4/4. You must multiply the whole number by the denominator first, then add the numerator.
- Changing the denominator during conversion is wrong because the size of each fractional piece stays the same. In 2 3/5 = 13/5, the denominator remains 5.
- Adding mixed numbers without using common denominators is wrong because fraction parts must represent equal-sized pieces before they can be combined. For example, 1/2 + 1/3 is not 2/5.
- Multiplying only the whole numbers and only the fractions is wrong because a mixed number is a sum, not two separate factors. Convert 2 1/3 × 1 1/2 to 7/3 × 3/2 before multiplying.
Practice Questions
- 1 Convert 4 3/7 to an improper fraction.
- 2 Compute 2 1/4 + 3 5/8 and write the answer as a mixed number in simplest form.
- 3 A student says 5 2/3 should be written as 7/3 because 5 + 2 = 7. Explain the error and give the correct improper fraction.