Fraction Calculator

Adding and subtracting fractions

Fractions can only be added when they share a denominator, because the denominator defines the size of the pieces you are counting. To add 1/2 and 1/3, convert both to sixths: 3/6 + 2/6 = 5/6.

The reliable method is cross-multiplication: a/b + c/d = (ad + cb) / bd. This always works, though it may not give the lowest common denominator, which is why the result should then be simplified. Subtraction follows the identical pattern with a minus sign.

Multiplying and dividing

Multiplication is the easy case: multiply numerators together and denominators together. 2/3 × 3/4 = 6/12, which simplifies to 1/2. No common denominator is needed.

Division means multiplying by the reciprocal — flip the second fraction and multiply. 1/2 ÷ 3/4 becomes 1/2 × 4/3 = 4/6 = 2/3. This works because dividing by 3/4 asks how many three-quarters fit into the first fraction, which is the same as multiplying by four-thirds.

Simplifying and mixed numbers

To reduce a fraction to lowest terms, divide the numerator and denominator by their greatest common divisor. For 6/12 the GCD is 6, giving 1/2. The Euclidean algorithm finds the GCD efficiently: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is zero.

An improper fraction, where the numerator exceeds the denominator, can be written as a mixed number. Divide to get the whole part and keep the remainder over the original denominator: 7/3 becomes 2 1/3. Improper fractions are usually easier to work with in calculations, while mixed numbers read more naturally in measurements and recipes. The calculator's mixed-number mode converts each to an improper fraction, operates, and converts back.

Converting between fractions and decimals

Dividing the numerator by the denominator gives the decimal. Fractions whose denominators have only 2 and 5 as prime factors terminate — 3/8 is exactly 0.375. Any other denominator produces a repeating decimal: 1/3 is 0.333… and 1/7 is 0.142857 repeating.

Going the other way, a terminating decimal becomes a fraction over a power of ten, then simplifies: 0.75 = 75/100 = 3/4. This is also why exact fraction arithmetic is preferable to decimals when precision matters — 1/3 has no exact decimal representation, so rounding errors accumulate in a way they do not with fractions.

Common fractions, decimals and percentages

Recognising these on sight saves a great deal of arithmetic. The ones with repeating decimals — thirds and sixths — are exactly the ones worth keeping as fractions.

Comparing, ordering and estimating fractions

To compare a/b with c/d (denominators positive), cross-multiply: compare a × d with c × b. For 1/2 and 2/3 the cross products are 3 and 4, and 3 < 4, so 1/2 < 2/3. Rewriting both over the least common denominator gives the same answer: 3/6 < 4/6.

To order a list, rewrite every fraction over the least common denominator and sort by numerator: 3/4, 2/3, 5/8 and 1/2 become 18/24, 16/24, 15/24 and 12/24, so the order is 1/2 < 5/8 < 2/3 < 3/4. Their average is the sum 61/24 divided by 4, which is 61/96.

For a quick estimate, round each fraction to the nearest benchmark 0, 1/2 or 1: 7/8 + 2/5 ≈ 1 + 1/2 = 1 1/2, while the exact sum is 51/40 = 1 11/40. The estimate is 9/40 high, close enough to catch a slip in the exact working.

Repeating decimals, percents and complex fractions

Long division turns any fraction into a decimal. If a remainder repeats, so do the digits after it: 1 ÷ 6 leaves remainder 4 again and again, so 1/6 = 0.1(6), also written 0.16 with a bar over the 6.

To reverse this for 0.1(6), let x = 0.1666…. Then 100x = 16.666… and 10x = 1.666…; subtracting cancels the tail, giving 90x = 15 and x = 15/90 = 1/6. A percent is a fraction over 100: 37.5% = 37.5/100 = 375/1000 = 3/8.

A complex fraction such as (3/4)/(5/6) has fractions in its numerator and denominator. Multiplying both by the least common denominator 12 clears them: 9/10. In a proportion a/b = c/d, cross-multiplying gives a × d = b × c, so 3/4 = 6/x gives x = 4 × 6 ÷ 3 = 8.

Fraction to decimal chart: halves to 64ths

Every 64th terminates because 64 = 2⁶, so each decimal below is exact. The lowest-terms column shows which ruler mark it is: 24/64 is 3/8 and 32/64 is 1/2. The Fraction table mode also lists every lowest-terms fraction from halves to sixteenths, including the repeating thirds, sixths and sevenths, or every n/d for one denominator up to 64.

The Number line mode draws halves through sixteenths on one 0-to-1 scale with decimal marks, and places your own fraction on it with the nearest eighth and sixteenth.