Add, subtract, multiply and divide fractions and mixed numbers. The result is shown as a simplified fraction, mixed number and decimal, with step-by-step working. You can also simplify a fraction or convert between a decimal and a fraction.
Simplified fraction
Decimal
Mixed number
Step-by-step working
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Simplified fraction
Decimal
Mixed number
Step-by-step working
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Simplified fraction
Mixed number
Step-by-step working
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A fraction represents a part of a whole and is written as the quotient of two integers. The numerator (top) shows how many parts you take and the denominator (bottom) how many equal parts the whole was split into. In 3/4 you take three of four parts: it equals 0.75, but in exact form.
A proper fraction has a numerator smaller than the denominator, so its value lies between 0 and 1. 1/2, 3/4 or 5/8 are proper: they do not make a whole.
If the numerator is greater than or equal to the denominator, the fraction is improper and is worth 1 or more (greater than or equal to one). 5/4, 7/3 or 8/8 can also be written as a mixed number or as a whole number.
A mixed number joins a whole number and a proper fraction: 1 3/4 is the same as 7/4. In this calculator the whole number sits to the left of the numerator; if you leave it empty, you work with the fraction only.
Two fractions are like fractions if they share a denominator (2/7 and 5/7) and unlike fractions if the denominators differ (1/2 and 1/3). To add or subtract unlike fractions you need a common denominator, almost always using the LCM.
An improper fraction and a mixed number express the same value in two forms. In class you are asked to convert from one to the other before operating, or to leave the result easier to read.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator; the denominator stays the same. 2 1/3 = (2 × 3 + 1)/3 = 7/3. In the form, that whole number is the field on the left.
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To go from an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole part and the remainder stays over the same denominator. 7/3 = 2 1/3, because 7 = 2 × 3 + 1.
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To add, subtract, multiply and divide fractions, the classroom method is to work only with fractions, proper or improper. If a mixed number appears, convert it first to an improper fraction: 2 1/3 = 7/3. Once everything is written as a/b you operate, simplify, and if the result is greater than 1 you can leave it improper or write it back as a mixed number.
To add fractions with the same denominator you do not need equivalent fractions: add the numerators and keep the denominator. 3/7 + 2/7 = 5/7. Simplify at the end if you can.
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If the denominators differ, the first step of the addition is to find a common denominator. The cleanest choice is the least common multiple (LCM) of both. Rewrite each fraction as an equivalent fraction with that denominator, then add the numerators.
You can also multiply the denominators together. It always works, but it usually gives a larger number than the LCM; after simplifying you reach the same result.
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Subtracting fractions follows the same rules as adding them: if needed, match the denominators with the LCM and then operate only on the numerators. The common denominator stays the same.
With the same denominator, subtract the numerators and keep the denominator. 5/7 − 2/7 = 3/7. If the GCD is not 1, reduce to lowest terms.
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With different denominators, find the LCM, write the equivalent fractions and subtract the numerators. 1/2 − 1/3 = 3/6 − 2/6 = 1/6.
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To multiply fractions you do not need a common denominator: multiply numerator by numerator and denominator by denominator. 1/2 × 2/3 = 2/6 = 1/3. Convert a mixed number to an improper fraction first.
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Dividing fractions is multiplying by the reciprocal of the second (flip and multiply): 2/3 becomes 3/2. So 1/2 ÷ 2/3 = 1/2 × 3/2 = 3/4.
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Simplifying a fraction (reducing it to lowest terms) does not change its value: divide the numerator and the denominator by their greatest common divisor (GCD) until it is in simplest form, which is what class asks for.
For 12/18, the GCD of 12 and 18 is 6, so 12/18 = 2/3. If the GCD is 1, the fraction was already simplified.
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To turn a decimal into a fraction, a terminating decimal is written over a power of 10 and simplified: 0.75 = 75/100 = 3/4. If it is greater than 1, it can also be written as a mixed number: 1.5 = 3/2 = 1 1/2. A repeating decimal such as 0.333… equals 1/3 (its generating fraction), not 333/1000.
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To turn a fraction into a decimal, divide the numerator by the denominator. 1/2 = 0.5 is a terminating decimal. If the decimal does not end, we show 6 digits and an ellipsis: 1/3 = 0.333333... and 1/7 = 0.142857.... Convert a mixed number to an improper fraction first: 2 1/3 = 7/3 = 2.333333....
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