Fractions Explained: Adding, Subtracting, Multiplying and Dividing

What a fraction is

A fraction represents a part of a whole. It has two numbers separated by a bar: the numerator on top tells you how many parts you have, and the denominator on the bottom tells you how many equal parts the whole was divided into. The fraction 3/8 means the whole was cut into 8 equal pieces and you have 3 of them.

Fractions come in three main types. A proper fraction has a smaller numerator than denominator, like 3/8, and is always less than 1. An improper fraction has a numerator equal to or larger than the denominator, like 11/4, and is worth 1 or more. A mixed number combines a whole number and a fraction, like 2 3/4. Improper fractions and mixed numbers hold the same value: 11/4 = 2 3/4, because 4 fits into 11 twice (that is 8 fourths) with 3 fourths left over.

Adding and subtracting fractions

You can only add or subtract fractions that share the same denominator — counting eighths and quarters together directly makes no more sense than adding hours to kilometers. If the denominators already match, just combine the numerators and keep the denominator. If they do not match, rewrite one or both fractions with a common denominator first.

To find a common denominator fast, multiply the two denominators together — it always works, even if it is not the smallest choice. For 3/4 and 1/6, using 4 × 6 = 24 also works: 3/4 = 18/24 and 1/6 = 4/24. The smaller denominator of 12 just keeps the numbers tidier.

Multiplying fractions

Multiplication is the easiest fraction operation — no common denominator needed. Multiply the two numerators, multiply the two denominators, then simplify the result. It helps to think of "2/3 of 3/5": you are taking a fraction of another fraction, which is exactly what multiplication means here.

A useful shortcut: before multiplying, cancel any factor that appears in a numerator and a denominator. For 2/3 × 3/5, the threes cancel, and you can read off the answer 2/5 directly.

Dividing fractions

Division uses one famous trick: flip the second fraction and multiply. The flipped version is called the reciprocal — 2/5 and 5/2 are reciprocals because their product is 10/10 = 1. Whole numbers are fractions too: 3 is 3/1, so its reciprocal is 1/3.

Why does flipping work? Dividing asks "how many of these fit into that?" Checking the answer confirms it: 3/4 ÷ 2/5 = 15/8, and multiplying back, 15/8 × 2/5 = 30/40 = 3/4. The reciprocal undoes the division, which is exactly what a correct answer must do.

Simplifying with the greatest common divisor

A fraction is in simplest form when nothing except 1 divides both the numerator and the denominator. To reduce a fraction, find the greatest common divisor (GCD) of the two numbers, then divide both by it. For 18 and 24, the GCD is 6: dividing gives 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so 18/24 = 3/4.

For big numbers, Euclid's algorithm finds the GCD without any factoring: divide the larger number by the smaller, then keep dividing the previous divisor by the remainder until the remainder is 0. The last nonzero remainder is the GCD.

Always simplify final answers. A result like 84/126 is correct but hard to picture; its simplified form tells the story at a glance.

Fractions, decimals and percents

Every fraction is a division in disguise — the fraction bar means "divided by." Dividing the numerator by the denominator turns a fraction into a decimal: 3 ÷ 8 = 0.375, so 3/8 = 0.375. Going the other way, a decimal becomes a fraction through place value: 0.625 reads as 625/1000, and dividing top and bottom by 125 gives 5/8.

Some fractions never land on a tidy decimal. 1/3 = 0.333… with the 3 repeating forever, and 5/6 = 0.8333… — these are called repeating decimals. In practice you round them to a few places. The common equivalents below are worth memorizing because they appear constantly in measurements, money and percentages.

Frequently asked questions