Formula Sheet • All 4 Operations & Conversions

Fractions Formula Sheet

Every fractions rule you need — adding, subtracting, multiplying, dividing, simplifying, mixed numbers, and converting to decimals and percentages.

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The Four Operations

Adding (Same Denominator)

\frac{a}{b} + \frac{c}{b} = \frac{a + c}{b}

Add the numerators together; keep the denominator unchanged.

Worked Example

\displaystyle \frac{2}{5} + \frac{1}{5} = \frac{2+1}{5} = \frac{3}{5}

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Adding (Different Denominators)

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Cross-multiply numerators and multiply denominators for a common denominator.

Worked Example

\displaystyle \frac{1}{3} + \frac{1}{4} = \frac{1(4) + 1(3)}{3 \times 4} = \frac{7}{12}

Fraction Calculator →

Subtracting

\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}

Find a common denominator, then subtract the cross-multiplied numerators.

Worked Example

\displaystyle \frac{3}{4} - \frac{1}{3} = \frac{3(3) - 1(4)}{4 \times 3} = \frac{5}{12}

Fraction Calculator →

Multiplying

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} = \frac{ac}{bd}

Multiply tops straight across; multiply bottoms straight across.

Worked Example

\displaystyle \frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}

Fraction Calculator →

Dividing (Keep • Change • Flip)

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

Keep the 1st fraction, change ÷ to ×, and flip the 2nd fraction.

Worked Example

\displaystyle \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}

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Reciprocal (Inverse)

\text{Reciprocal of } \frac{a}{b} = \frac{b}{a}

Flip numerator and denominator. Their product always equals 1.

Worked Example

\displaystyle \frac{3}{4} \times \frac{4}{3} = \frac{12}{12} = 1

Fraction Calculator →

Simplifying & Equivalent Fractions

Simplify to Lowest Terms

\frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)}

Divide both numerator and denominator by their HCF / GCD.

Worked Example

\displaystyle \gcd(12, 18) = 6 \implies \frac{12 \div 6}{18 \div 6} = \frac{2}{3}

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Equivalent Fractions (Scale Up)

\frac{a}{b} = \frac{a \times n}{b \times n} \quad (n \neq 0)

Multiply numerator and denominator by the exact same whole number n.

Worked Example

\displaystyle \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}

Equivalent Fractions →

Equivalence Test (Cross-Multiply)

\frac{a}{b} = \frac{c}{d} \iff ad = bc

Two fractions are equal if and only if cross-products are equal.

Worked Example

\displaystyle \frac{2}{5} \stackrel{?}{=} \frac{6}{15} \implies 2(15) = 5(6) = 30 \;\checkmark

Equivalent Fractions →

Mixed Numbers & Improper Fractions

Mixed Number to Improper

a\,\frac{b}{c} = \frac{(a \times c) + b}{c}

Multiply whole number a by denominator c, then add numerator b.

Worked Example

\displaystyle 2\,\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{11}{4}

Mixed Number Calculator →

Improper to Mixed Number

\frac{a}{b} = Q\,\frac{R}{b} \quad (a = Q \times b + R)

Divide a ÷ b: quotient Q is whole part, remainder R is new numerator.

Worked Example

\displaystyle \frac{11}{4} = 11 \div 4 = 2 \text{ r } 3 \implies 2\,\frac{3}{4}

Mixed Number Calculator →

Adding Mixed Numbers

A\,\frac{b}{d} + C\,\frac{e}{d} = (A+C)\,\frac{b+e}{d}

Add whole numbers together, then add fraction parts with common denominators.

Worked Example

\displaystyle 2\,\frac{1}{4} + 1\,\frac{2}{4} = (2+1) + \frac{1+2}{4} = 3\,\frac{3}{4}

Mixed Number Calculator →

Fraction of an Amount

Find Fraction of an Amount

\frac{a}{b} \times N = (N \div b) \times a

Divide the total amount N by denominator b, then multiply by numerator a.

Worked Example

\displaystyle \frac{3}{4} \text{ of } 40 = (40 \div 4) \times 3 = 30

Fraction Calculator →

Find Whole from Known Part

\text{Whole} = (\text{Part} \div a) \times b

Divide the part by numerator a to get 1 unit, then multiply by denominator b.

Worked Example

\displaystyle \text{If } \frac{3}{4} = 30 \implies (30 \div 3) \times 4 = 40

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Part as Fraction of Whole

\text{Fraction} = \frac{\text{Part}}{\text{Whole}}

Ensure both quantities share identical units, then simplify the fraction.

Worked Example

\displaystyle 45\text{ min of } 1\text{ hr} = \frac{45}{60} = \frac{3}{4}

Fraction Calculator →

Converting Fractions

Fraction to Decimal

\frac{a}{b} = a \div b

Divide the numerator by the denominator using short division or a calculator.

Worked Example

\displaystyle \frac{3}{8} = 3 \div 8 = 0.375

Fraction to Decimal Converter →

Fraction to Percentage

\left(\frac{a}{b}\right) \times 100\% = (a \div b) \times 100\%

Convert the fraction to a decimal first, then multiply by 100.

Worked Example

\displaystyle \frac{3}{8} = 0.375 \times 100\% = 37.5\%

Equivalent Fractions →

Decimal to Fraction

0.d_1 d_2 = \frac{d_1 d_2}{100} = \text{simplify}

Place digits over 10, 100, or 1000 depending on decimal places, then divide by HCF.

Worked Example

\displaystyle 0.75 = \frac{75}{100} = \frac{75 \div 25}{100 \div 25} = \frac{3}{4}

Fraction to Decimal Converter →

Curriculum & Revision Note

About These Fractions Formulas

Fractions underpin a huge proportion of the school maths curriculum — appearing throughout algebra, ratio, probability, and trigonometry. This sheet covers all four arithmetic operations, equivalence, simplifying, mixed numbers, and unit conversions.

Need step-by-step solutions? Open Fraction Calculator →