SamCalculator

Fraction Calculator

Add, subtract, multiply, and divide fractions with steps.

Fractions

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What Is a Fraction?

A fraction represents a part of a whole, written as a numerator over a denominator separated by a horizontal bar. The denominator tells you how many equal pieces the whole has been divided into; the numerator counts how many of those pieces you have. The fraction 3/4 means three of four equal parts, and every rational number — every value that either terminates or repeats in decimal form — can be expressed exactly as a fraction.

This Fraction Calculator handles the four basic operations — addition, subtraction, multiplication, and division — on two fractions, with full step-by-step solutions, automatic reduction to lowest terms, and a visual pie chart comparison. Pair it with our percentage calculator, ratio calculator, or common factor calculator for related rational-number work.

How Fraction Operations Work

Common denominators for + and −

Find the least common multiple of the two denominators, scale each fraction up so both share that LCM as the bottom number, then add or subtract the numerators while keeping the denominator unchanged.

Straight-across for ×

Multiply the two numerators to get the new numerator, multiply the two denominators to get the new denominator. No common-denominator step required.

Reciprocal flip for ÷

Dividing by a fraction is the same as multiplying by its reciprocal. Flip the second fraction (n/d becomes d/n), then multiply straight across.

Reduce to lowest terms

After any operation, divide both numerator and denominator by their greatest common factor. That step is always automatic in this calculator.

6 Ways to Use This Calculator

1

Homework checking

Verify any fraction arithmetic answer — the step-by-step explanation shows whether you forgot a common denominator or a final reduction.

2

Recipe scaling

Multiply 2/3 cup of flour by 1.5 to triple-half a recipe — exact arithmetic without rounding error.

3

Measurement work

Add 3/8 + 9/16 in a single tap to get 15/16 — no decimal rounding for carpentry and machining.

4

Probability calculations

Combine event probabilities like 1/6 × 1/6 = 1/36 for double-six odds, kept in exact reduced form.

5

Algebra worksheets

Solve fraction subproblems mid-equation and keep every intermediate result exact, never decimal.

6

Quick mental-math sanity check

Confirm a mental estimate against an exact answer before committing to a longer derivation.

Best Practices

Always reduce when finishing. 35/180 is technically correct, but 7/36 is the canonical form and the answer most teachers expect. This calculator reduces automatically, but learning to do it by hand prevents marks lost on worksheets.

Keep numbers as fractions through long chains. 1/3 in decimal is 0.333… repeating; rounding to 0.333 and chaining several multiplications compounds error. Stay in fraction form until the very last conversion.

Sanity-check with rough decimals. A correct fraction answer should be close to your decimal mental estimate. If 7/8 + 1/3 comes out near 1.21, you're in the right neighbourhood; if it comes out near 8, you misplaced a denominator.

Why Fractions Matter

School & exams

Fractions underpin elementary, middle, and high-school math. Confident fraction work unlocks algebra, functions, calculus, and beyond.

Engineering & trades

Inch fractions are the lingua franca of carpentry, plumbing, and machining in the US. Even metric shops see fractional inches on drawings.

Pure mathematics

Rational numbers — fractions of integers — are the cleanest closed system under +, −, ×, and ÷, and the foundation of number theory.

Daily life

Recipes, money, sports scores, music notation, and time-of-day all use fractions because they communicate exact parts of a whole naturally.

Tricky Cases & Edge Behaviour

Negative fractions

The minus sign settles on the numerator after simplification; −3/4 and 3/−4 are the same value. The calculator always outputs the canonical positive-denominator form.

Zero in the denominator

Division by zero is undefined, so the calculator rejects inputs with a zero denominator before computing rather than returning infinity.

Improper results

Adding 5/6 + 7/6 gives 12/6, which reduces to 2 — a whole number. The mixed-number form is shown alongside the improper form.

Very small denominators

Operations like 1/1000000 × 1000000 reduce exactly to 1, with no floating-point error, because every step uses exact BigInt arithmetic.

Core Fraction Formulas

Every answer this calculator produces comes from one of these identities. Let a/b and c/d be the operands and g be the greatest common factor of the unreduced result.

Addition

a/b + c/d = (a·d + b·c) / (b·d)

Subtraction

a/b − c/d = (a·d − b·c) / (b·d)

Multiplication

a/b × c/d = (a·c) / (b·d)

Division

a/b ÷ c/d = (a·d) / (b·c)

Simplification

(a/b) / g = (a/g) / (b/g)

Decimal

a/b = a ÷ b

Percentage

a/b × 100% = (a × 100)/b %

Reciprocal

1 ÷ (a/b) = b/a

Common Fraction Mistakes

Adding numerators and denominators

1/2 + 1/3 is NOT 2/5. Add only the numerators after rewriting with a common denominator: 3/6 + 2/6 = 5/6.

Forgetting to reduce

Leaving 6/8 as the final answer is technically correct but loses style points and hides the underlying value of 3/4.

Flipping the wrong fraction in ÷

When dividing a/b ÷ c/d, flip the SECOND fraction (c/d → d/c) and multiply. Flipping both leaves the value unchanged.

Cross-multiplying for + and −

Cross-multiplication is for solving proportions (a/b = c/d), not for combining fractions. Use the LCM method for addition and subtraction.

Cancelling within a sum

In (a + b)/c you cannot cancel the c with anything inside the sum until the addition completes. The cancel-as-you-go shortcut works only across multiplication.

Decimal rounding traps

Computing 1/3 as 0.333 then adding three of them gives 0.999, not the exact 1. Stay in fraction form to keep results exact.

Built for students, teachers, engineers, machinists, and anyone who needs exact rational arithmetic in the browser.

Every step uses exact BigInt arithmetic — no floating-point rounding anywhere in the pipeline. See our methodology and editorial policy.

Frequently Asked Questions

A fraction represents a part of a whole. It is written as one integer (the numerator) over another integer (the denominator), separated by a horizontal bar — for example 3/4 means three of four equal parts. Fractions with a numerator smaller than the denominator are called proper fractions; ones where the numerator is larger are improper fractions; and a whole number combined with a proper fraction is a mixed number.

Find the least common multiple (LCM) of the denominators, multiply each fraction so both share that common denominator, then add the numerators while keeping the denominator unchanged. Finally, reduce the result to lowest terms using the greatest common factor. For example, 2/7 + 3/8: the LCM of 7 and 8 is 56, so 2/7 = 16/56 and 3/8 = 21/56, giving 37/56.

Subtraction follows the same recipe as addition: convert both fractions to the LCM of the denominators, subtract the numerators, then reduce. The sign of the result follows the larger absolute value. The calculator handles negative inputs automatically.

Multiply the two numerators together to get the new numerator, multiply the two denominators together to get the new denominator, then reduce. For 5/12 × 7/15 you get 35/180, which reduces to 7/36 by dividing both top and bottom by the GCD of 5.

Dividing by a fraction is the same as multiplying by its reciprocal. To compute a/b ÷ c/d, flip the second fraction to d/c and multiply: (a × d)/(b × c). Then reduce. The calculator validates that the divisor is non-zero before computing.

A proper fraction has a numerator smaller than its denominator (e.g. 3/4). An improper fraction has a numerator equal to or larger than its denominator (e.g. 7/4). A mixed number combines a whole part with a proper fraction (e.g. 1 3/4). Every improper fraction can be rewritten as a mixed number and vice versa.

Find the greatest common factor (GCF) of the numerator and denominator, then divide both by it. For 84/126, the GCF is 42, so the simplified form is 2/3. The calculator displays the GCF and the division steps.

Count the digits after the decimal point, write the number without the decimal as the numerator, and write 10 raised to that digit count as the denominator. Then reduce. For example, 1.375 = 1375/1000 = 11/8 = 1 3/8 in mixed form.

Divide the numerator by the denominator using long division. The calculator shows both the terminating decimal (when the denominator's prime factors are only 2 and 5) and any repeating cycle for other fractions — for example 2/7 = 0.285714285714… repeating every six digits.

The big-number tab uses JavaScript BigInt for the underlying arithmetic, so it is exact regardless of input size. There is no precision loss when adding, subtracting, multiplying, or dividing fractions whose numerator and denominator are hundreds of digits long.

Related Calculators

Pair the fraction calculator with other math tools for ratios, percentages, factorisation, and step-by-step number work.