Fraction

Calculating with Fractions

Adding, subtracting, multiplying, or dividing two fractions combines them into a single simplified result. Enter two fractions and choose an operation, and this calculator returns the simplified result, its decimal equivalent, and a mixed-number form when the result is an improper fraction (numerator larger than denominator). Each fraction also has an optional whole-number field, so a mixed number like 1 1/2 can be entered directly instead of converting it to the improper fraction 3/2 first.

A second mode, “Convert a decimal to a fraction,” goes the other direction: enter any decimal and get back its simplest equivalent fraction — useful for turning a measurement or a calculator’s decimal output into a fraction you can actually mark on a tape measure or a recipe.

The Formula

Adding or subtracting first finds a common denominator by cross-multiplying:

$$\frac{\vA{a}}{\vB{b}} + \frac{\vC{c}}{\vD{d}} = \frac{\vA{a} \times \vD{d} + \vC{c} \times \vB{b}}{\vB{b} \times \vD{d}}$$

Multiplying multiplies straight across:

$$\frac{\vA{a}}{\vB{b}} \times \frac{\vC{c}}{\vD{d}} = \frac{\vA{a} \times \vC{c}}{\vB{b} \times \vD{d}}$$

Dividing multiplies by the second fraction’s reciprocal:

$$\frac{\vA{a}}{\vB{b}} \div \frac{\vC{c}}{\vD{d}} = \frac{\vA{a} \times \vD{d}}{\vB{b} \times \vC{c}}$$

Every result is then reduced to its simplest form by dividing both the numerator and denominator by their greatest common divisor.

Converting a decimal to a fraction uses a continued-fraction expansion — the standard method for finding the simplest fraction that closely approximates a given decimal, rather than just reading off digits after the decimal point (which would turn a repeating decimal like 0.333333 into an ugly fraction like 333333/1000000 instead of the actual 1/3 it’s approximating).

Worked Example

Adding 1/2 and 1/3:

1. Common denominator: $2 \times 3 = 6$.
2. Numerator: $1 \times 3 + 1 \times 2 = 5$.
3. Result: $\frac{5}{6}$ (already in simplest form) — about 0.8333.

Adding 1/2 and 2/3 instead gives an improper fraction: $\frac{7}{6}$, which this calculator also shows as the mixed number 1 1/6.

Key Factors to Consider

• Simplifying to lowest terms always gives an equivalent value, not just a “cleaner-looking” number. Reducing 5/10 to 1/2 doesn’t change what fraction of a whole it represents — both describe exactly the same quantity, just expressed with smaller numbers, which is why the simplified form is preferred by convention rather than necessity.
• Cross-multiplying to add or subtract works for any two fractions, but isn’t always the most efficient method. Finding the least common denominator (rather than simply multiplying the two denominators together, which this calculator’s cross-multiply approach does before simplifying) can sometimes involve smaller intermediate numbers — but both methods always converge on the same final, simplified answer.
• A negative fraction follows the same rules, with the sign tracked through each operation. Multiplying or dividing fractions with different signs (one positive, one negative) produces a negative result, following the same sign rules as regular multiplication and division.
• Dividing by a fraction is the same as multiplying by its reciprocal, a rule worth remembering beyond just this calculator. This “flip and multiply” shortcut for fraction division comes directly from the definition of division as the inverse of multiplication, and applies whether you’re dividing by hand or using a tool like this one.

Common Mistakes

• Adding or subtracting fractions by combining the numerators and denominators directly. As covered above, fractions need a common denominator before the numerators can be combined — adding 1/2 + 1/3 as (1+1)/(2+3) gives the wrong answer (2/5) instead of the correct 5/6.
• Forgetting to simplify the final result. As covered above, an unsimplified answer like 4/8 is mathematically correct but not in the conventional simplest form (1/2) most contexts expect — always reduce by the greatest common divisor before treating a result as final.
• Flipping the wrong fraction when dividing. As covered above, dividing by a fraction means multiplying by that fraction’s reciprocal — flipping the first fraction instead of the second produces the reciprocal of the correct answer, not the answer itself.
• Converting a decimal to a fraction by simply reading off the digits after the decimal point. As covered above, this works for a terminating decimal but produces an unnecessarily large, unsimplified fraction for a repeating one — 0.333333 read literally becomes 333333/1000000 instead of the actual 1/3 it’s approximating.

Useful to Know

• A mixed number and its improper-fraction form always represent the same value — 1 1/2 and 3/2 are two different ways of writing an identical quantity, useful for different contexts (a recipe reads more naturally as a mixed number, while further math is often easier with an improper fraction).
• The greatest common divisor used to simplify a fraction is the same GCD calculated by the GCF/LCM Calculator — worth checking there if you want to see that step worked out on its own.
• This calculator pairs naturally with the Ratio Calculator (for comparing two quantities rather than combining them) and the Long Division Calculator (for the division process a fraction’s decimal equivalent comes from).