Basic Fraction Formulas with Worked Operations

Understand numerator, denominator and equivalence

In a/b, the numerator a counts parts and the nonzero denominator b specifies the division. Multiplying both by the same nonzero number preserves the value: 2/3 = 10/15 because both entries were multiplied by five. Dividing both by a shared factor also preserves the value, so 18/24 = 3/4 after division by six. Simplification does not mean making only the numerator smaller. Changing one entry alone generally changes the number, even if the new fraction looks easier to read.

Use equal-sized parts for addition and subtraction

With a common denominator, a/b + c/b = (a+c)/b and a/b − c/b = (a−c)/b. For different denominators, rewrite the fractions first. Thus 3/4 + 1/6 = 9/12 + 2/12 = 11/12, while 3/4 − 1/6 = 7/12. These operations count or remove pieces of a common size. Keeping the original denominators and adding them together would produce an unrelated quantity, so the denominator rule differs from multiplication.

Multiply numerators and denominators

The product is (a/b)(c/d) = ac/(bd). For 4/9 multiplied by 3/8, the direct result is 12/72 = 1/6. Cancelling shared factors before multiplication gives the same answer with smaller intermediate values: cancel three between 3 and 9, then four between 4 and 8. Such cancellation applies to factors in a product. It does not authorize crossing out terms across a plus sign, where addition binds the terms into a different expression.

Divide by multiplying by the reciprocal

For c nonzero, (a/b) ÷ (c/d) = ad/(bc). The reciprocal belongs to the divisor, not the first fraction. For 5/6 ÷ 10/9, calculate 5/6 × 9/10 = 45/60 = 3/4. A divisor of 0/7 would be zero and is invalid even though its denominator is allowed. As a check, multiply the quotient 3/4 by the original divisor 10/9 to recover 5/6; this reverses the division and confirms the result.

Convert mixed numbers before applying the formulas

A positive mixed number w n/d means w + n/d = (wd+n)/d. Therefore 2 2/5 becomes 12/5. The usual negative mixed notation −2 2/5 means the negative of the complete quantity, giving −12/5, rather than −2 + 2/5. Convert before entering the separate numerator and denominator boxes. When several operations appear in one written expression, evaluate parentheses first, then multiplication and division, then addition and subtraction; a two-fraction form performs only the single operation selected for that run.

Keep an exact fraction until the final display

A fraction and a rounded decimal are different representations of a result. For example, 1/3 is exact, while 0.333333 is a six-place approximation. Converting that rounded decimal back into a fraction gives 333333/1000000, not exactly 1/3. When continuing a calculation, carry the reduced fraction into the next step instead of copying a short decimal preview. The local Fraction Calculator accepts integer numerators and denominators and performs the four basic operations on those rational values. Its decimal line is labelled as rounded, so use the fraction line when an exact answer is required.

Check the denominator and the size of the answer

A denominator cannot be zero: a/b means a divided by b, and division by zero has no value in ordinary fraction arithmetic. A zero numerator with a nonzero denominator is valid and simplifies to zero. Negative values are also valid; keep the sign attached to the complete fraction when moving between notations. Before accepting a result, estimate its size. Adding two positive fractions should give something larger than either addend, whereas multiplying a positive number by a fraction between zero and one makes it smaller. Such checks can catch a wrong operation even when the calculator has evaluated the entered numbers correctly.

Compare without rounding

To compare 5/8 and 7/12, use a common denominator of 24: the fractions become 15/24 and 14/24. Thus 5/8 is larger. Comparing only the numerators would suggest the opposite and ignores the different sizes of the pieces.

Related guides

Fraction Addition Formula: Common Denominators

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Calculate a Fraction of an Amount

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Convert an Improper Fraction to a Mixed Number

Use quotient and remainder to convert an improper fraction, simplify its fractional part, and interpret negative mixed numbers consistently.