Fraction calculator

How to add and subtract fractions by hand

Fractions can only be added or subtracted once they count the same size of piece, so the first job is a common denominator. Work through 5/12 + 7/18:

1. Find the least common denominator, the LCM of 12 and 18. Count up in multiples of the larger denominator until one divides by the smaller: 18, 36. Since 36 ÷ 12 = 3, the LCD is 36.
2. Scale each fraction up to it: 5/12 = 15/36 (top and bottom × 3) and 7/18 = 14/36 (× 2).
3. Add the numerators and keep the denominator: 15/36 + 14/36 = 29/36.
4. Simplify. The GCD of 29 and 36 is 1, so 29/36 is already in lowest terms.

The calculator returns 29/36 for these inputs, or 0.80(5) as a decimal. Two other routes to the LCD give the same 36: multiply the denominators and divide by their GCD (12 × 18 ÷ 6 = 216 ÷ 6), or take each prime factor at its highest power (12 = 2² × 3 and 18 = 2 × 3², so 2² × 3² = 36). The GCD and LCM calculator shows both for larger numbers.

Subtracting mixed numbers has one extra decision. For 3 1/4 − 1 5/6, either convert both to improper fractions first (13/4 − 11/6 = 39/12 − 22/12 = 17/12 = 1 5/12), or keep the whole numbers apart and borrow: over twelfths the subtraction is 3 3/12 − 1 10/12, and because 3/12 is smaller than 10/12, borrow 1 from the 3 to make 2 15/12. Then 2 15/12 − 1 10/12 = 1 5/12. Both methods agree; converting first avoids the borrowing step and is what the calculator does.

Multiplying and dividing fractions

Multiplication needs no common denominator: multiply the numerators, multiply the denominators, then simplify. Cancelling shared factors diagonally before multiplying keeps the numbers small. In 8/15 × 25/32, the 8 and the 32 share a factor of 8, and the 25 and the 15 share 5, which leaves 1/3 × 5/4 = 5/12. Multiplying straight through gives 200/480, and dividing by the GCD, 40, reaches the same 5/12; the calculator's steps show this second route.

A whole number is a fraction over 1, so 4 × 2/3 = 4/1 × 2/3 = 8/3, or 2 2/3. Scaling a recipe works the same way: making one and a half batches turns 2/3 cup of flour into 2/3 × 3/2 = 1 cup, and 3/4 teaspoon of salt into 9/8 = 1 1/8 teaspoons.

Division turns into multiplication by the reciprocal of the second fraction, the divisor. With mixed numbers, convert first: 3/4 ÷ 1 1/2 = 3/4 ÷ 3/2 = 3/4 × 2/3 = 6/12 = 1/2. Reading a division as "how many of these fit into that" is a quick check: 2 1/2 ÷ 1 1/4 = 5/2 × 4/5 = 2, and two pieces of 1 1/4 do make 2 1/2.

Simplifying fractions with the greatest common divisor

A fraction is in lowest terms when its numerator and denominator have no common factor above 1. Dividing both by their greatest common divisor gets there in one step. For small numbers the GCD can be spotted; for large ones, Euclid's algorithm (Elements, Book VII, Proposition 2) finds it by repeated division, which is also how the calculator does it. To simplify 462/1071:

1. 1071 = 2 × 462 + 147
2. 462 = 3 × 147 + 21
3. 147 = 7 × 21 + 0

The last non-zero remainder, 21, is the GCD. Dividing top and bottom by 21 gives 462/1071 = 22/51. Cancelling small primes in turn works here too, by 3 (154/357) and then by 7 (22/51), because both numbers happen to have small factors; Euclid's method does not depend on that.

Mixed numbers and improper fractions

A proper fraction has a numerator smaller than its denominator (3/8), an improper fraction does not (47/6), and a mixed number writes an improper fraction as a whole number plus a proper fraction (7 5/6). To turn an improper fraction into a mixed number, divide with remainder: 47 ÷ 6 = 7 remainder 5, so 47/6 = 7 5/6. The calculator shows both forms for any improper result.

Negative mixed numbers catch people out. −1 3/4 means −(1 + 3/4) = −7/4, not −1 + 3/4 = −1/4. The minus sign applies to the whole number and the fraction together, and the calculator reads "-1 3/4" that way.

Fraction to decimal chart

Inch measurements in halves, quarters, eighths and sixteenths all end within four decimal places. A fraction in lowest terms whose denominator is 2^a × 5^b ends after the larger of a and b places: 3/16 has 16 = 2⁴, so it needs four (0.1875), and 7/40 has 40 = 2³ × 5, so it needs three (0.175).

Denominators with any other prime factor repeat. Brackets mark the block that repeats, the same notation the calculator accepts as input.

Common fraction mistakes

• Adding across. 1/2 + 1/3 is not 2/5. Over the common denominator 6 it is 3/6 + 2/6 = 5/6. A sum of two positive fractions is always larger than each of them, and 2/5 is smaller than 1/2.
• Cancelling terms instead of factors. In (2 + 3)/2 the 2s cannot be struck out to leave 3; the value is 5/2. Cancelling works only on numbers that multiply the whole numerator and the whole denominator.
• Flipping the wrong fraction. In a division, only the divisor, the second fraction, is inverted.
• Comparing by the denominator alone. To compare 5/8 and 3/5, cross-multiply: 5 × 5 = 25 against 3 × 8 = 24, so 5/8 is larger, by exactly 1/40.
• Typing a rounded decimal. The calculator treats 0.33 as exactly 33/100. For one third, type 1/3 or the repeating form 0.(3).

To express a fraction as a share of 100, the percentage calculator takes the numerator and denominator as X and Y. Fractions that compare two quantities, such as 3 parts sand to 1 part cement, are often easier to scale with the ratio calculator.