How Fractions Work

A fraction is just a division problem written as two stacked numbers — and that one idea covers recipes scaled up or down, guitar string lengths, half of all "2/3 cup" cooking, and the decimal numbers you use every day.

The anatomy of a fraction

Every fraction has two parts. The denominator (bottom) says how many equal pieces the whole was cut into; the numerator (top) says how many of those pieces you have. So 3/4 means: cut into 4 equal pieces, take 3 of them.

The bar between them is a division sign. 3/4 is 3 ÷ 4 = 0.75. Every fraction is a number, and every fraction has a decimal twin — which is why "one half" and "0.5" and "50%" are three costumes on the same value.

Fractions also come in infinitely many disguises for the same amount: 1/2 = 2/4 = 6/12 = 50/100. Multiplying (or dividing) both parts by the same number never changes the value, because you're effectively multiplying by 1: 2/4 is just 1/2 with both parts doubled.

Simplifying with the GCD

The standard form of a fraction is the one where numerator and denominator share no common factor. To find it, divide both by their greatest common divisor (GCD). Take 84/126:

GCD(84, 126) = 42
84 ÷ 42 = 2,  126 ÷ 42 = 3
84/126 = 2/3

Why bother? Two reasons. First, it's easier to reason with 2/3 than with 84/126. Second, simplest form makes comparison possible: is 3/8 bigger than 5/12? Cross-multiply (3×12 = 36 vs 5×8 = 40) and you can see 3/8 is smaller — a comparison that's awkward until both fractions are in clean form.

The four operations

Addition and subtraction need a common denominator, because you can only add pieces of the same size — adding thirds to quarters is like adding meters to feet. Find the least common denominator and convert both fractions:

1/3 + 1/4 = 4/12 + 3/12 = 7/12

Multiplication needs no such step — multiply straight across, tops and bottoms separately:

2/3 × 3/5 = (2×3)/(3×5) = 6/15 = 2/5

Notice how the common-denominator rule connects to the decimal percentage world: 1/3 + 1/4 ≈ 0.333 + 0.250 = 0.583, and 7/12 ≈ 0.583. Same answer, two languages. (For more on that translation, see how to calculate percentages.)

Division means "how many of the second fit into the first," and the rule is to multiply by the reciprocal — flip the second fraction upside down:

2/3 ÷ 3/5 = 2/3 × 5/3 = 10/9 = 1 1/9

Converting decimals to fractions

A terminating decimal is a fraction whose denominator is a power of ten — the digits after the point are the numerator, and the number of digits sets the denominator:

0.375 = 375/1000 = 3/8   (divide both by 125)

Reducing afterward is essential; 375/1000 and 3/8 are the same number, but only 3/8 is useful. Common memorables: 0.5 = 1/2, 0.25 = 1/4, 0.125 = 1/8, 0.6 = 3/5, 0.625 = 5/8, 0.75 = 3/4.

Repeating decimals are fractions too, just ones that never terminate: 1/3 = 0.333…, and 1/7 = 0.142857142857…, a six-digit pattern that repeats forever. In fact every repeating decimal is exactly some fraction — that's a defining property of rational numbers, and it's why "rational" literally means "ratio."

Improper fractions and mixed numbers

When the numerator is at least the denominator — like 7/4 — the fraction is called improper, though there's nothing wrong with it. The friendlier costume is the mixed number: 7/4 = 1 whole with 3/4 left over = 1 3/4. Converting back and forth is simple arithmetic: 7 ÷ 4 gives quotient 1 remainder 3.

Mixed numbers read naturally in everyday contexts (recipe amounts, heights, carpenter's measurements), while improper fractions are much easier to compute with. A good habit: convert to improper fractions, do the math, then convert the answer back.

Try it yourself

The Fraction Calculator adds, subtracts, multiplies, and divides fractions and shows the result in simplest form. The Fraction Simplifier reduces any fraction (or finds the GCD) instantly, and the Decimal to Fraction Converter translates any decimal — including repeating ones — into its fraction form.