Fraction calculator

How to write mixed numbers and repeating decimals

Write a mixed number, also called a mixed fraction, with a space between the whole number and the fraction: 2 1/2. Put only the repeating digits of a decimal in brackets: 0.(3) is 1/3, and 0.1(6), which is 0.1666…, is 1/6.

• A minus sign covers the whole mixed number. -1 3/4 means −(1 + 3/4) = −7/4, not −1 + 3/4 = −1/4.
• Bars and dots become brackets. Textbooks often mark the repeating block with a bar or dots above it; in brackets, 1/7 is 0.(142857).
• Terminating decimals are exact. 2.375 is exactly 2375/1000, which reduces to 19/8, or 2 3/8.
• Irrational numbers have no exact fraction. π and √2 never terminate or repeat, so a fraction such as 22/7 only approximates π.

How to add, subtract, multiply and divide fractions

Add or subtract fractions over a common denominator: 1/4 + 2/3 = 3/12 + 8/12 = 11/12. Multiply straight across: 2/3 × 3/4 = 6/12 = 1/2. To divide, multiply by the reciprocal of the second fraction: 3/4 ÷ 1/8 = 3/4 × 8/1 = 6. Finally, simplify by dividing top and bottom by their greatest common divisor.

The four rules in symbols, for fractions a/b and c/d:

• Add or subtract: a/b + c/d = (ad + bc)/bd. Using the least common denominator instead of bd keeps the numbers smaller.
• Multiply: a/b × c/d = ac/bd.
• Divide: a/b ÷ c/d = a/b × d/c = ad/bc, which needs c ≠ 0.

Adding fractions with different denominators

Fractions can only be combined once they count the same size of piece. 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.

Two other routes reach the same LCD of 36: divide the product of the denominators by their GCD (12 × 18 ÷ 6), or take each prime factor at its highest power (2² × 3 and 2 × 3² give 2² × 3²).

The GCD and LCM calculator finds both values for larger denominators.

Mixed numbers add one decision. For 3 1/4 − 1 5/6, the quickest route converts both to improper fractions first: 13/4 − 11/6 = 39/12 − 22/12 = 17/12 = 1 5/12.

Keeping the whole numbers apart needs borrowing instead: 3 3/12 − 1 10/12 becomes 2 15/12 − 1 10/12 = 1 5/12.

Multiplying and dividing fractions

Multiplication needs no common denominator. Cancel shared factors diagonally first to keep the numbers small: in 8/15 × 25/32, the 8 and 32 share 8 and the 25 and 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.

A whole number is a fraction over 1: 3/5 × 7 = 3/5 × 7/1 = 21/5, or 4 1/5. Scaling a recipe by one and a half 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.

To divide, flip the divisor and multiply. The reciprocal of 5/13 is 13/5, so dividing by 5/13 means multiplying by 13/5. Convert mixed numbers first: 3/4 ÷ 1 1/2 = 3/4 ÷ 3/2 = 3/4 × 2/3 = 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.

How to simplify a fraction

A fraction is in lowest terms when its numerator and denominator share no factor above 1. Dividing both by their greatest common divisor gets there in one step.

For large numbers, Euclid's algorithm (Elements, Book VII, Proposition 2) finds the GCD by repeated division. To simplify 462/1071:

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

The last nonzero remainder, 21, is the GCD, so 462/1071 = 22/51. Canceling small primes in turn also works here, by 3 (154/357) and then by 7, but only because both numbers happen to have small factors.

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 it 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. To go back, multiply the whole number by the denominator and add the numerator: (7 × 6 + 5)/6 = 47/6.

How to convert a decimal to a fraction

A terminating decimal is a fraction over a power of ten. Write the digits over 10, 100 or 1,000 and simplify: 0.375 = 375/1000 = 3/8, and 13.5 = 135/10 = 27/2, or 13 1/2.

A repeating decimal takes one subtraction. For x = 0.666…, 10x = 6.666…, so 10x − x = 6, which gives 9x = 6 and x = 2/3.

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.

So 3/16, with 16 = 2⁴, needs four places (0.1875), and 7/40, with 40 = 2³ × 5, needs three (0.175).

Denominators with any other prime factor give a decimal that repeats forever. Brackets mark the repeating block: 0.(09) means 0.090909….

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 larger than each of them, and 2/5 is smaller than 1/2.
• Canceling terms instead of factors. In (2 + 3)/2 the 2s cannot be struck out to leave 3; the value is 5/2. Canceling 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. 0.33 is exactly 33/100, not one third. For one third, use 1/3 or the repeating form 0.(3).

To express a fraction as a share of 100, the percentage calculator takes the numerator as the part and the denominator as the whole.

Fractions that compare two quantities, such as 3 parts sand to 1 part cement, are often easier to scale with the ratio calculator.