Fractions, Decimals & Percents formulas

All 20 formulas in this topic

Adding Fractionsab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}Add fractions with different denominators by finding a common denominator, then combining the numerators to get your answer.Subtracting Fractionsab−cd=ad−bcbd\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}Subtracting fractions with different denominators requires a common denominator to combine the numerators into a single fraction.Multiplying Fractionsab⋅cd=acbd\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}Multiply fractions by multiplying numerators and denominators separately, letting you combine fractional amounts and solve fraction problems efficiently.Dividing Fractionsab÷cd=ab⋅dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}Dividing fractions means flipping the second fraction and multiplying instead — it lets you split quantities into equal groups.Simplifying Fractionsab=a÷gcd⁡(a,b)b÷gcd⁡(a,b)\frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)}Simplify a fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common factor.Equivalent Fractionsab=akbk,k≠0\frac{a}{b} = \frac{ak}{bk}, \quad k \neq 0Multiply or divide both parts of a fraction by the same non-zero number to create an equivalent fraction with the same value.Comparing Fractionsab>cd  ⟺  ad>bc(b,d>0)\frac{a}{b} > \frac{c}{d} \iff ad > bc \quad (b, d > 0)Compare two fractions to find which is larger or smaller without finding a common denominator—use when you need a quick size comparison.Mixed Number to Improper Fractionwab=wb+abw\frac{a}{b} = \frac{wb + a}{b}Convert a mixed number into an improper fraction so you can multiply, divide, or compare it with other fractions in calculations.Reciprocalab⋅ba=1\frac{a}{b} \cdot \frac{b}{a} = 1A reciprocal flips a fraction—multiply a number by its reciprocal and you always get 1, helping you solve fraction equations.Complex Fractions;ab;cd=adbc\frac{;\frac{a}{b};}{\frac{c}{d}} = \frac{ad}{bc}Learn to simplify complex fractions where the numerator and denominator are themselves fractions using the multiply-by-reciprocal rule.Fraction to Decimalab=a÷b\frac{a}{b} = a \div bConvert a fraction to its decimal form by dividing the numerator by the denominator, useful for comparing fractions directly.Repeating Decimal to Fraction0.d1⋯dn‾=d1⋯dn10n−10.\overline{d_1 \cdots d_n} = \frac{d_1 \cdots d_n}{10^n - 1}Convert repeating decimals to fractions when you need to express infinite decimals like 0.333... as exact fractions for calculation and comparison.Fraction to Percentp%=ab×100%p\% = \frac{a}{b} \times 100\%Converting a fraction to a percentage tells you what proportion out of 100 you represent, making it easier to compare quantities on a standard scale.Percent of a Numberpart=p100⋅w\text{part} = \frac{p}{100} \cdot wPercent of a Number finds how much you get when taking a percentage of a quantity—essential for tips, discounts, grades, and everyday calculations.Percent Change% change=new−oldold×100%\%\,\text{change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%Percent change calculates how much a quantity has increased or decreased, expressed as a percentage of its original value.Percent Error% error=∣measured−actualactual∣×100%\%\,\text{error} = \left| \frac{\text{measured} - \text{actual}}{\text{actual}} \right| \times 100\%Percent Error measures how far off a measurement is from the true value, as a percentage — use it to check accuracy of lab work or approximations.Discount and Sale PriceS=L(1−d100)S = L\left(1 - \frac{d}{100}\right)Calculate the sale price after subtracting a discount percentage from the original list price of the item you're buying.Sales Tax and TipT=P(1+r100)T = P\left(1 + \frac{r}{100}\right)Calculate the total final amount you pay when adding a percentage—such as sales tax or tip—to the original price of any item.Simple InterestI=PrtI = PrtSimple Interest calculates the total earnings or charges on borrowed or invested money using principal, rate, and time, without compounding effects.Compound InterestA=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}Compound Interest calculates how money grows when interest earns interest, used for investments, savings, and loans over time.

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