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Fraction Formulas for Add, Subtract, Multiply, Divide

Review exact fraction formulas for mixed-number conversion, common denominators, multiplication, reciprocal division, reduction, decimals, and percentages.

How do the Fraction Calculator formulas work?

This formula page defines the variables used by the Fraction Calculator, shows how those values are combined, and explains the order of operations behind the result. Use the worked steps to verify an estimate, compare scenarios, and understand which assumptions or rounding rules can affect the displayed answer.

Variable Definitions

Let a/b be the first fraction and c/d be the second. The denominators b and d must be nonzero integers. Let w represent a whole-number part, n a numerator, and g the greatest common factor used in reduction.

Mixed Number to Improper Fraction

w  nd=wd+ndw\;\frac{n}{d}=\frac{wd+n}{d}

For a negative mixed number, the sign applies to the entire quantity before arithmetic begins.

Add and Subtract Fractions

ab±cd=ad±bcbd\frac{a}{b}\pm\frac{c}{d}=\frac{ad\pm bc}{bd}

The direct formula is valid for nonzero denominators. The calculator uses the least common multiple as the working denominator when possible, which keeps intermediate values smaller without changing the exact answer.

Multiply and Divide Fractions

ab×cd=acbd\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}
ab÷cd=ab×dc=adbc,c≠0\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}=\frac{ad}{bc},\quad c\ne0

Division is undefined when the second fraction equals zero because its reciprocal would have a zero denominator.

Reduce the Result to Lowest Terms

g=gcd⁡(∣n∣,d),nd=n/gd/gg=\gcd(|n|,d),\qquad \frac{n}{d}=\frac{n/g}{d/g}

Dividing both terms by their greatest common factor preserves the value and produces the unique reduced fraction with a positive denominator.

Convert to Decimal and Percentage

x=nd,p=100×nd%x=\frac{n}{d},\qquad p=100\times\frac{n}{d}\%

The exact rational answer is retained for the fraction and mixed-number forms. Decimal and percentage displays are rounded to two decimal places when necessary.

Worked Calculation

23+34=812+912=1712=1  512\frac{2}{3}+\frac{3}{4}=\frac{8}{12}+\frac{9}{12}=\frac{17}{12}=1\;\frac{5}{12}

The least common denominator is 12. The result is already reduced because 17 and 12 have no common factor greater than 1.

References

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