How fraction arithmetic works
A fraction has two parts. The numerator on top counts how many pieces you have. The denominator on the bottom says how many equal pieces make one whole. So 3/4 means three pieces out of four. The calculator keeps every answer as an exact fraction, then shows it as a mixed number and a decimal as well.
Adding and subtracting
You can only add or subtract pieces of the same size, so the denominators must match first. The smallest shared denominator is the least common multiple (LCM) of the two denominators.
Example: 1/4 + 1/6. The LCM of 4 and 6 is 12. Rewrite 1/4 as 3/12 and 1/6 as 2/12. Then 3/12 + 2/12 = 5/12. Subtraction works the same way: 3/4 − 1/3 = 9/12 − 4/12 = 5/12.
You could also just multiply the denominators (4 × 6 = 24). That gives 6/24 + 4/24 = 10/24, which is right but has to be simplified at the end. Using the LCM keeps the numbers smaller.
Multiplying
Multiply the tops together and the bottoms together. No common denominator is needed.
Example: 2/3 × 3/4 = 6/12, which simplifies to 1/2.
Dividing
Dividing by a fraction is the same as multiplying by its reciprocal (the fraction flipped upside down). Teachers often call this “keep, change, flip”.
Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8, which is 1 7/8 as a mixed number. You cannot divide by zero, so the second fraction must not be 0.
Mixed numbers
A mixed number such as 2 3/4 is a whole number plus a fraction. To calculate with it, turn it into an improper fraction first: multiply the whole number by the denominator and add the numerator.
So 2 3/4 = (2 × 4 + 3)/4 = 11/4. Type it in the box with a space: 2 3/4. A minus sign in front applies to the whole thing, so -2 3/4 is −11/4, not −2 + 3/4.
Worked example: 1 1/2 + 2 3/4. Convert: 3/2 + 11/4. The LCM of 2 and 4 is 4, so 6/4 + 11/4 = 17/4. As a mixed number that is 4 1/4, and as a decimal 4.25.
Simplifying a fraction
A fraction is in lowest terms when the top and bottom share no factor except 1. To get there, divide both by their greatest common divisor (GCD). The calculator finds the GCD with Euclid’s algorithm: divide the bigger number by the smaller, keep the remainder, and repeat until the remainder is 0. The last non-zero remainder is the GCD.
Example: simplify 18/24. 24 = 1 × 18 + 6, then 18 = 3 × 6 + 0. The GCD is 6, so 18/24 = (18 ÷ 6)/(24 ÷ 6) = 3/4.
Euclid’s method is fast even for huge numbers, which is why the calculator can simplify fractions with 30-digit numerators exactly.
Fractions, decimals and percentages
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.33% |
| 1/4 | 0.25 | 25% |
| 1/6 | 0.1666… | 16.67% |
| 3/8 | 0.375 | 37.5% |
| 2/3 | 0.666… | 66.67% |
| 7/8 | 0.875 | 87.5% |
A fraction gives a decimal that ends only when its denominator, in lowest terms, has no prime factors other than 2 and 5. That is why 3/8 = 0.375 stops, but 1/3 and 1/7 repeat forever. The calculator shows repeating decimals with the repeating part in brackets: 1/6 = 0.1(6) and 1/7 = 0.(142857).
Going the other way, any decimal you type is turned into an exact fraction. 0.375 is 375/1000, and dividing by the GCD of 125 gives 3/8.
Tips and common mistakes
- Do not add the denominators. 1/2 + 1/3 is not 2/5. Use a common denominator: 3/6 + 2/6 = 5/6.
- Simplify before you multiply if the numbers are big. In 8/9 × 3/4, cancel the 8 with the 4 and the 3 with the 9 first to get 2/3 straight away.
- Watch the sign. A negative divided by a negative is positive: −3/4 ÷ −2/5 = 15/8.
- Check with a decimal. If 1/4 + 1/6 = 5/12, then 0.25 + 0.1667 should be about 0.4167, and it is.
For percentage questions, such as what share 5/12 is of the whole, use the percentage calculator. To find how much a value has grown in percent, try the percentage increase calculator. For powers, roots and trigonometry, open the scientific calculator.
All calculations happen in your browser with exact whole-number arithmetic, so there are no rounding errors in the fraction answer, and nothing you type is sent anywhere.