Fractions Explained: Types and How to Calculate Them

A fraction is a way to write part of a whole, using a top number over a bottom number. This page covers what fractions are, the three fraction types, and how to add, subtract, multiply and divide them.

What is a fraction?

A fraction is a number written as a quotient, one whole number divided by another (Britannica). It has two parts: the numerator on top and the denominator on the bottom.

In the fraction 1/3, the numerator is 1 and the denominator is 3. The denominator tells you how many equal pieces a whole is cut into, and the numerator tells you how many of those pieces you have. So 1/3 means one piece out of three equal pieces, and you can read it as “one-third.”

This section covers whole-number fractions only. It doesn’t cover fractions with variables or negative numbers, which come up later in algebra.

What are proper, improper and mixed fractions?

A proper fraction has a numerator smaller than its denominator, like 3/4. An improper fraction has a numerator that is greater than or equal to its denominator, like 7/4 or 5/5.

A mixed number combines a whole number with a proper fraction, like 1 3/4. You can rewrite any improper fraction as a mixed number, and any mixed number as an improper fraction.

Improper fraction to mixed number. Take 11/4. Divide the numerator by the denominator: 11 divided by 4 goes 2 times, with 3 left over. The quotient (2) becomes the whole number, and the remainder (3) becomes the new numerator over the same denominator. So 11/4 equals 2 3/4. Check it: 2 wholes plus 3/4 is the same amount as 11 fourths, since 2 x 4 + 3 = 11.

Mixed number to improper fraction. Take 2 1/3. Multiply the whole number by the denominator: 2 x 3 = 6. Add the numerator: 6 + 1 = 7. Keep the same denominator. So 2 1/3 equals 7/3. Check it: 7 divided by 3 goes 2 times with 1 left over, which brings you back to 2 1/3.

This section covers converting between the two forms only. It doesn’t cover comparing improper fractions to decimals, which belongs in a decimals lesson.

What are equivalent fractions, and how do you simplify a fraction?

Equivalent fractions are different fractions that name the same amount, like 1/2 and 2/4. You get an equivalent fraction by multiplying or dividing both the numerator and denominator by the same nonzero number.

Worked example. Is 6/8 equivalent to 3/4? Divide both numbers in 6/8 by their greatest common factor, 2: 6 ÷ 2 = 3, and 8 ÷ 2 = 4. That gives 3/4, so yes, 6/8 and 3/4 are equivalent.

Simplifying, also called reducing, means dividing the numerator and denominator by their greatest common factor until the fraction can’t be divided down any further. A fraction is fully simplified when the only number that divides both parts evenly is 1.

Worked example. Simplify 8/12. The greatest common factor of 8 and 12 is 4. Divide both by 4: 8 ÷ 4 = 2, and 12 ÷ 4 = 3. So 8/12 simplifies to 2/3.

This section covers fractions with whole-number numerators and denominators. It doesn’t cover simplifying algebraic fractions or ratios, which use related but separate rules (see our ratios and proportions page).

How do you add and subtract fractions?

To add or subtract fractions, the denominators must match first. If they already match, add or subtract the numerators and keep the denominator the same.

Same denominator. 1/6 + 3/6. Add the numerators: 1 + 3 = 4. Keep the denominator: 4/6. Simplify by dividing both by 2: 4/6 = 2/3.

If the denominators are different, find a common denominator first, usually the least common multiple of the two denominators, then rewrite each fraction using it.

Different denominators. 1/4 + 1/3. The least common multiple of 4 and 3 is 12. Rewrite 1/4 as 3/12 (multiply top and bottom by 3) and 1/3 as 4/12 (multiply top and bottom by 4). Add the numerators: 3/12 + 4/12 = 7/12.

Subtracting example. 3/4 − 1/6. The least common multiple of 4 and 6 is 12. Rewrite 3/4 as 9/12 and 1/6 as 2/12. Subtract: 9/12 − 2/12 = 7/12.

This section covers adding and subtracting two fractions. It doesn’t cover adding fractions with mixed numbers directly, though the same method works once you convert any mixed numbers to improper fractions first.

How do you multiply and divide fractions?

To multiply fractions, multiply the numerators together and multiply the denominators together, then simplify. There’s no need to find a common denominator first.

Multiplying example. 2/3 x 3/5. Multiply the numerators: 2 x 3 = 6. Multiply the denominators: 3 x 5 = 15. That gives 6/15, which simplifies to 2/5 (divide both by 3).

To divide by a fraction, flip the second fraction (swap its numerator and denominator) and multiply instead. This flipped fraction is called the reciprocal.

Dividing example. 3/4 ÷ 2/5. Flip 2/5 to get its reciprocal, 5/2. Multiply: 3/4 x 5/2. Multiply the numerators: 3 x 5 = 15. Multiply the denominators: 4 x 2 = 8. The answer is 15/8, or 1 7/8 as a mixed number.

This section covers multiplying and dividing fractions with whole-number numerators and denominators. It doesn’t cover complex fractions or fractions with unknowns, which come up in algebra.

Practice set

Work through these on paper, then check your answers.

1. 1/2 + 1/4
2. 2/3 − 1/6
3. 3/5 x 2/3
4. 5/6 ÷ 1/3
5. Simplify 6/8
6. Convert the mixed number 3 2/5 to an improper fraction
7. Convert the improper fraction 13/5 to a mixed number

Fractions show up everywhere once you start looking, from splitting a pizza to reading a tape measure. They’re also the base skill behind percentages and behind solving fraction word problems. If you want to put fractions to work right away, try scaling a recipe up or down using the same multiplying and dividing rules from this page. For more math topics, visit our math hub.

Sources

• Britannica. Fraction.