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Add, subtract, multiply, and divide fractions easily with simplified output values.
Working with fractions: the four operations and why only one needs a common denominator
A fraction is a division waiting to happen. The numerator on top is how many parts you have, the denominator underneath is how many parts make a whole, and the line between them means divide. Most of the difficulty people have with fractions comes from a single asymmetry: adding and subtracting need a common denominator, while multiplying and dividing do not.
The four operations
| Operation | Rule | Example |
|---|---|---|
| Add | Common denominator, then add numerators | 2/3 + 1/4 = 8/12 + 3/12 = 11/12 |
| Subtract | Common denominator, then subtract numerators | 3/4 − 1/6 = 9/12 − 2/12 = 7/12 |
| Multiply | Numerators together, denominators together | 2/3 × 3/5 = 6/15 = 2/5 |
| Divide | Flip the second fraction and multiply | 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6 |
Multiplication needs no common denominator because you are taking a part of a part, which does not require the pieces to be the same size. Addition does, because you cannot combine thirds and quarters until both are expressed in twelfths.
Finding the common denominator
Any common multiple works, and the product of the two denominators always is one. Using the lowest common multiple just keeps the numbers smaller. For 1/6 + 1/8, the product gives forty-eighths and the LCM gives twenty-fourths; both reach 7/24 after simplifying, but the second route avoids the extra step.
LCM(a, b) = a × b / GCD(a, b)
The greatest common divisor is found with the Euclidean algorithm: divide, keep the remainder, repeat until the remainder is zero. For 48 and 18: 48 ÷ 18 leaves 12, 18 ÷ 12 leaves 6, 12 ÷ 6 leaves 0, so the GCD is 6.
Simplifying
Divide both numerator and denominator by their GCD. 18/24 has a GCD of 6, giving 3/4. A fraction is in lowest terms when the only number dividing both parts is 1. The calculator does this automatically and also returns the mixed number and decimal forms.
Mixed numbers and improper fractions
An improper fraction has a numerator at least as large as its denominator. To convert 17/5 to a mixed number, divide: 17 ÷ 5 is 3 remainder 2, so 3 2/5. Going back, multiply the whole number by the denominator and add the numerator: 3 × 5 + 2 = 17, giving 17/5. Do all arithmetic in improper form and convert at the end; mixed numbers are for reading, not for calculating.
Fractions, decimals and the ones that never end
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.33% |
| 1/4 | 0.25 | 25% |
| 2/3 | 0.666… | 66.67% |
| 1/8 | 0.125 | 12.5% |
| 5/6 | 0.8333… | 83.33% |
| 7/16 | 0.4375 | 43.75% |
A fraction terminates as a decimal only when its denominator, in lowest terms, has no prime factors other than 2 and 5. That is why halves, quarters, fifths, eighths and tenths are tidy and thirds, sixths, sevenths and ninths are not. It is a property of base ten, not of the fractions: in base twelve, thirds terminate neatly.
A worked example
A recipe for four needs 2/3 cup of flour and you are cooking for six. Multiply by 6/4, which is 3/2: (2 × 3) / (3 × 2) = 6/6 = 1 cup exactly. If it needed 3/4 cup instead, 3/4 × 3/2 = 9/8 = 1 1/8 cups.
Frequently Asked Questions (FAQs)
Why do I need a common denominator to add but not to multiply?
Because adding combines quantities, which requires the pieces to be the same size, while multiplying takes a portion of a portion, which does not. Half of a third is a sixth regardless of how the pieces are labelled, so the denominators can simply be multiplied.
How do I divide one fraction by another?
Multiply by the reciprocal: flip the second fraction and multiply. Dividing by 3/4 is the same as multiplying by 4/3. It works because dividing by a number is defined as multiplying by its inverse, and the inverse of a fraction is that fraction turned over.
What is the fastest way to simplify a large fraction?
Find the greatest common divisor with the Euclidean algorithm and divide both parts by it once. Repeatedly halving or dividing by small primes also works but takes more steps. For 144/180 the GCD is 36, giving 4/5 in one move.
How do I convert a repeating decimal back to a fraction?
For a single repeating digit, put it over 9: 0.777… is 7/9. For two repeating digits, put them over 99: 0.4545… is 45/99, which simplifies to 5/11. Non-repeating digits before the cycle need a slightly longer method involving subtraction.
Can a fraction have a negative denominator?
Mathematically yes, but by convention the sign is moved to the numerator or in front of the whole fraction. −3/4, 3/−4 and −(3/4) are the same number, and the first form is standard.
Why does 0.999... equal 1?
Because 1/3 is 0.333… and multiplying both sides by three gives 1 = 0.999…. They are two notations for the same number, not two numbers that are very close. The infinite expansion leaves no gap between them.