Math · 5 min read
How to Work with Fractions
Add, subtract, multiply, divide — then reduce by the gcd
A fraction a/b is a ratio of integers with a nonzero denominator. This tutorial walks through the four arithmetic operations, a 3/9 + 1/6 worked example that reduces to 1/2, and a playground with two fractions, an operation toggle, and unit bars. The Fraction Calculator on this site uses the same identities.
Written by the My Calculator Stack editorial team. About our methods
Denominators must be nonzero. Results are reduced by the greatest common divisor. This is arithmetic on rational numbers, not mixed-number formatting for a gradebook.
1.What a fraction is
The numerator a counts parts; the denominator b says how many parts make a whole. Equivalent fractions name the same rational number: 3/9 and 1/3 are the same point on the number line because both sides divide by 3. Operations combine two fractions into one new numerator and denominator, then you reduce by dividing out the greatest common divisor (gcd).
2.The formulas
Addition and subtraction need a common denominator: a/b ± c/d = (ad ± bc) / (bd). Multiplication is straight across: (a/b)×(c/d) = ac/bd. Division multiplies by the reciprocal: (a/b)÷(c/d) = ad/bc, so the second numerator must be nonzero. After any operation, divide numerator and denominator by gcd(|n|, |d|).
\[ \begin{aligned}\tfrac{a}{b}\pm\tfrac{c}{d}&=\tfrac{ad\pm bc}{bd} \\\tfrac{a}{b}\times\tfrac{c}{d}&=\tfrac{ac}{bd} \\\tfrac{a}{b}\div\tfrac{c}{d}&=\tfrac{ad}{bc}\end{aligned} \]
3.Worked example
Add 3/9 + 1/6 — the same pair the Fraction Calculator starts with. Common-denominator form is (3×6 + 9×1) / (9×6) = (18 + 9) / 54 = 27/54. gcd(27, 54) = 27, so 27/54 reduces to 1/2. You can also simplify 3/9 to 1/3 first: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2.
\[ \tfrac{3}{9}+\tfrac{1}{6}=\tfrac{1}{2} \]
4.Two-fraction playground
Set two numerators and denominators, then choose add, subtract, multiply, or divide. The unit bars fill to each fraction and to the reduced result; the caption substitutes the live numbers. This is a teaching widget — no favorites, no full calculator chrome. Open the Fraction Calculator when you want typed values and a decimal too.
Inline playground
First fraction a / b
Second fraction c / d
Reduced result
1/2
3/9 + 1/6 = 1/2
a/b = 3/9
c/d = 1/6
Result = 1/2
Teaching aid only — denominators must be nonzero. Division by a zero numerator is undefined.
Open the full Fraction calculator →5.Open the full calculator
The Fraction Calculator uses the same four operations plus a simplify-only mode, the same 3/9 and 1/6 defaults, and How it works plus FAQ on one page. Browse other math calculators if you need percentages, rounding, or equations next.
Try it yourself
Open the related calculator and put these formulas to work.