Fraction Calculator

A fraction describes part of a whole. The bottom number, the denominator, says how many equal pieces the whole is cut into, and the top number, the numerator, says how many of those pieces you have. Three eighths, written 3/8, is three pieces of a whole that was cut into eight. When the numerator is smaller than the denominator the fraction is called proper and is worth less than one; when it is equal or larger, as in 9/4, the fraction is improper and can also be written as a mixed number, 2 1/4.

Many different fractions name the same amount. Cutting a pizza into 8 slices and eating 4 is the same as cutting it into 2 and eating 1, so 4/8 = 1/2. Multiplying or dividing the numerator and denominator by the same non-zero number never changes the value, and that single idea drives almost every fraction technique: finding a common denominator, simplifying, and converting to decimals.

Addition:

Fractions can only be added directly when they count pieces of the same size, which means they need the same denominator. If they already share one, add the numerators and keep the denominator: 2/9 + 5/9 = 7/9.

When the denominators differ, rewrite both fractions over a common denominator first. The quickest way that always works is to multiply each fraction's top and bottom by the other fraction's denominator. For a/b + c/d this gives (a×d + c×b)/(b×d). With the calculator's default values, 2/7 + 3/8 = (2×8 + 3×7)/(7×8) = (16 + 21)/56 = 37/56. Because 37 is prime and does not divide 56, the answer is already in lowest terms.

Multiplying the denominators together can produce large numbers. A tidier method is to use the least common multiple of the denominators, also called the least common denominator. To add 5/12 + 1/18, notice that the smallest number both 12 and 18 divide into is 36. Scale each fraction to that denominator: 5/12 = 15/36 and 1/18 = 2/36. Then 15/36 + 2/36 = 17/36. The cross-multiplication method gives 5×18 + 1×12 = 102 over 216, which reduces to the same 17/36, but only after an extra simplifying step.

With three or more fractions, either add them two at a time or bring them all to one common denominator at once. For 1/2 + 1/3 + 1/4 the least common denominator is 12, so the sum is 6/12 + 4/12 + 3/12 = 13/12, or 1 1/12 as a mixed number.

Mixed numbers can be added by converting each to an improper fraction first. 1 2/3 becomes (1×3 + 2)/3 = 5/3. Alternatively, add the whole parts and the fractional parts separately, then carry any extra whole that the fractions produce.

Subtraction:

Subtraction works exactly like addition except that the numerators are subtracted. Bring both fractions to a common denominator, subtract the tops, and simplify: a/b − c/d = (a×d − c×b)/(b×d). For example 3/4 − 1/6 = (18 − 4)/24 = 14/24 = 7/12. If the second fraction is larger the result is negative, as in 1/5 − 1/2 = (2 − 5)/10 = −3/10.

Multiplication:

Multiplying fractions is the easiest operation because no common denominator is needed. Multiply the numerators together and the denominators together: a/b × c/d = (a×c)/(b×d). So 2/3 × 5/8 = 10/24, which simplifies to 5/12.

You can save work by cancelling before you multiply. Any factor shared by a numerator and either denominator can be divided out first. In 2/3 × 5/8, the 2 and the 8 share a factor of 2, leaving 1/3 × 5/4 = 5/12 directly. To multiply a fraction by a whole number, treat the whole number as having a denominator of 1.

Division:

Dividing by a fraction is the same as multiplying by its reciprocal, the fraction turned upside down. To divide a/b by c/d, multiply a/b by d/c: (a×d)/(b×c). For instance 3/5 ÷ 9/10 = 3/5 × 10/9 = 30/45 = 2/3.

This rule makes sense if you ask how many times the divisor fits into the dividend. Dividing by 1/4 asks how many quarters fit, and there are four quarters in every whole, so dividing by 1/4 multiplies by 4. Division by zero is undefined, so a fraction with a numerator of zero cannot be used as a divisor. When dividing mixed numbers, convert them to improper fractions first; dividing the whole and fractional parts separately does not give the right answer.

Simplification:

A fraction is in simplest form, or lowest terms, when the numerator and denominator have no common factor other than 1. To simplify, find the greatest common factor of the two numbers and divide both by it. For 21/98 the greatest common factor is 7, so 21/98 = 3/14.

If the greatest common factor is not obvious, divide out small primes one at a time. 84/126 halves to 42/63, then dividing both by 3 twice gives 14/21 and then by 7 gives 2/3. The Euclidean algorithm finds the greatest common factor directly for larger numbers.

Converting between fractions and decimals:

To turn a fraction into a decimal, divide the numerator by the denominator. 3/8 = 3 ÷ 8 = 0.375. A fraction in lowest terms gives a terminating decimal only when its denominator has no prime factors other than 2 and 5. Any other denominator produces a repeating decimal: 1/3 = 0.333…, and 2/7 = 0.285714285714…, where the block 285714 repeats forever. The fraction-to-decimal tool marks the repeating block in brackets.

To turn a terminating decimal into a fraction, write the digits over the matching power of ten and simplify. A decimal with three places sits over 1,000: 1.375 = 1375/1000. The greatest common factor of 1375 and 1000 is 125, so the fraction reduces to 11/8, or 1 3/8.

Repeating decimals convert with a little algebra. Let x = 0.272727…, where 27 repeats. Multiplying by 100 shifts the decimal two places: 100x = 27.2727…. Subtracting the original removes the repeating tail, leaving 99x = 27, so x = 27/99 = 3/11. The power of ten matches the length of the repeating block.

Frequently asked questions