Fraction Calculator

Operations with fractions

The fraction calculator performs four basic operations and automaticallyreduces the result to the irreducible formand displays it as a mixed number and a decimal fraction.

How do you calculate fractions?

• addition/subtraction:reduce to a common denominator, add/subtract numerators,
• multiplication:numerator × numerator, denominator × denominator,
• division:multiply by the reciprocal of the second fraction.

Example: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Operations with fractions - rules in a nutshell

• Addition/subtraction: reduceto a common denominator (preferably NWW), add numerators: 1/4 + 1/6 = 3/12 + 2/12 =5/12.
• Multiplication:numerator × numerator, denominator × denominator (cross-reduce before multiplying!): 2/3 × 9/10 = 18/30 =3/5.
• Division:multiply by the inverse: 3/4 ÷ 2/5 = 3/4 × 5/2 =15/8 = 1⅞.
• Reducing:divide numerator and denominator by GCD: 24/36 → GCD=12 →2/3.
• Comparing:common denominator or cross multiplication: 5/7 vs 3/4 → 20 vs 21 → 3/4 bigger.

The calculator performs all operations on common and mixed fractions,showing the step-by-step solution of- common denominator, reduction and the result as a fraction and decimal.

Fractions in life - not only at school

Kitchen recipes (¾ cup of flour when converting to 1.5 servings - i.e. ¾ × 3/2 = 9/8 = 1⅛ cup), inch measurements in DIY (5/16″ wrench, 3/4″ plywood), division of a bill or plot of land. You can convert a fraction into a percentage by dividing the numerator by the denominator (3/8 = 0.375 = 37.5%) - thepercentage calculatoris also used for this purpose, and GCD and NWW can be found in the NWW/GCD calculator.

Fraction and decimal expansion - when the expansion ends

An irreducible fraction hasfinite decimal expansion only when its denominator is decomposed only into twos and fives(form 2ᵃ·5ᵇ): 3/8 = 0.375 (because 8 = 2³), 7/20 = 0.35 (20 = 2² 5), 9/40 = 0.225. Any other denominator givesperiodic expansion: 1/3 = 0.(3), 5/6 = 0.8(3), 2/7 = 0.(285714) - the digits in brackets repeat indefinitely. Therefore, 1/3 cannot be written "exactly" as a decimal number, and 0.33 is only an approximation. When ordering fractions on the number line, the easiest way is to convert them all to decimals: 2/5 = 0.4 < 5/12 ≈ 0.417 < 3/7 ≈ 0.429.

Common student mistakes - check before handing in

• Adding numerators and denominators:1/2 + 1/3 is NOT 2/5! First, the common denominator: 3/6 + 2/6 = 5/6. A quick sanity test: the sum of two positive fractions cannot be less than the larger component - and 2/5 < 1/2.
• Division without inversion:3/4 ÷ 1/2 = 3/4 × 2/1 = 3/2, not 3/8.
• "A larger denominator = a larger fraction":is the opposite - 1/3 > 1/4, because we divide the whole into fewer parts.
• Reducing by subtraction:from 24/36 you cannot "subtract 4" - we divide the numerator and denominator (by GCD = 12, i.e. 2/3).
• Losing the whole in mixed numbers:2½ × 3 = 7½ (because 2½ = 5/2), not 6½.

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