How the four operations work on fractions
A fraction names equal parts of one whole: the bottom number (denominator) says how many parts the whole is cut into, and the top number (numerator) says how many of those parts you have. Each of the four operations handles that pair of numbers in its own way.
Adding and subtracting only work when both fractions use parts of the same size. One third and one quarter cannot be added as they stand, because thirds are bigger than quarters. The fix is to cut both into a size that fits each of them, here twelfths: 1/3 is 4/12 and 1/4 is 3/12, and four twelfths plus three twelfths is seven twelfths.
Multiplying needs no common denominator. Taking 2/3 of 9/10 means multiplying the tops together and the bottoms together, which gives 18/30.
Dividing asks how many times the second fraction fits into the first. Turning the divisor upside down (its reciprocal) and multiplying gives the same answer, so 3/4 ÷ 1/2 becomes 3/4 × 2/1 = 6/4.
The calculator keeps every number as a whole-number numerator and denominator from start to finish, so 1/3 stays exactly one third instead of becoming 0.333 partway through.
Fraction formulas
Each result is then divided top and bottom by their greatest common factor, so it ends up in lowest terms. Dividing by a fraction whose numerator is 0 means dividing by zero, which has no answer.
Multiplying the two denominators always produces a common denominator, though not always the smallest one. Using the least common multiple of the denominators keeps the numbers smaller by hand; either route lands on the same simplified answer.
Lowest terms and mixed numbers
A fraction is in lowest terms when its numerator and denominator share no factor except 1. To get there, divide both by their greatest common factor (GCF). For 18/30 the GCF is 6, and 18 ÷ 6 over 30 ÷ 6 is 3/5. Dividing by a smaller shared factor, such as 2, leaves 9/15, which still reduces.
A mixed number like 2 1/2 is a whole number plus a proper fraction. The calculator accepts mixed numbers with a space between the whole part and the fraction, as well as whole numbers and terminating decimals such as 0.25, and turns each into an exact fraction before calculating:
- Mixed to improper: multiply the whole number by the denominator and add the numerator. 3 5/8 is (3 × 8 + 5)/8 = 29/8.
- Improper to mixed: divide the numerator by the denominator; the quotient is the whole part and the remainder is the new numerator. 17/4 is 4 remainder 1, so 4 1/4.
A negative mixed number is negative as a whole: −1 1/2 means −(1 + 1/2) = −3/2, not −1 + 1/2.
Worked examples
- The least common multiple of 6 and 10 is 30, so use thirtieths.
- 5/6 = 25/30 and 7/10 = 21/30.
- Add the numerators: 25 + 21 = 46, giving 46/30.
- The GCF of 46 and 30 is 2, so 46/30 = 23/15.
- As a mixed number, 23 ÷ 15 is 1 remainder 8: 1 8/15. The decimal 1.5(3) repeats the 3 forever.
- Convert to improper fractions: 2 1/2 = 5/2 and 1 3/4 = 7/4.
- Rewrite 5/2 as 10/4 so both are in quarters.
- Subtract: 10/4 − 7/4 = 3/4.
- Convert: 1 2/3 = 5/3 and 2 1/4 = 9/4.
- Multiply across: (5 × 9)/(3 × 4) = 45/12.
- The GCF of 45 and 12 is 3, so 45/12 = 15/4, which is 3 3/4.
- Write 1 1/2 as 3/2.
- Flip the divisor and multiply: 3/2 × 8/3 = 24/6.
- 24/6 simplifies to 4, a whole number.
Where fraction answers go wrong
- Adding the denominators. 1/3 + 1/4 is not 2/7. The denominators set the size of the parts, so they are matched first, never added.
- Multiplying mixed numbers piece by piece. 1 2/3 × 2 1/4 is not 1 × 2 plus 2/3 × 1/4 (which gives 2 1/6). Convert to improper fractions first; the correct answer is 3 3/4.
- Flipping the wrong fraction when dividing. Only the second fraction, the divisor, is inverted. Flipping the first one gives the reciprocal of the right answer.
- Stopping at a common factor that is not the greatest. Dividing 24/36 by 2 gives 12/18, which still reduces. Dividing by the GCF, 12, gives 2/3 in one step.
- Rounding to a decimal too early. Working with 0.33 for 1/3 and then multiplying by 3 gives 0.99, not 1. Keep fractions until the last step, which is what the calculator does.
The decimal under each result comes from exact long division, not from rounding. When it repeats, the repeating digits are shown in parentheses, so 0.58(3) means 0.58333… with the 3 repeating. A repeating block longer than the display is cut short and marked with an ellipsis.
Questions people ask
Can I type a negative fraction?
Yes. Put a minus sign in front, as in −3/4 or −1 1/2. The sign applies to the whole number, including a mixed number’s fractional part.
Why is my answer an improper fraction and a mixed number?
They are the same value written two ways. Improper fractions such as 23/15 are easier to keep calculating with, while mixed numbers such as 1 8/15 are easier to picture, for example on a measuring cup or a ruler.
Sources
- Fraction, Wolfram MathWorld.
- Mixed Fraction, Wolfram MathWorld.
- Euclid's algorithm, Dictionary of Algorithms and Data Structures, National Institute of Standards and Technology.