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Arithmetic & Basics

Everyday number sense: fractions, percentages, ratios and averages

Percentage

Basic
%=partwhole×100\% = \frac{\text{part}}{\text{whole}} \times 100

What fraction of a whole a part represents, out of 100.

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Percent of a Number

Basic
value=p100×N\text{value} = \frac{p}{100} \times N

Find p percent of a number N.

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Percentage Change

Basic
%change=new−oldold×100\% \text{change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100

Increase or decrease of a value as a percentage.

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Fraction to Decimal

Basic
ab=a÷b\frac{a}{b} = a \div b

Convert a fraction into a decimal by dividing.

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Adding Fractions

Basic
ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Add two fractions using a common denominator.

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Multiplying Fractions

Basic
ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Multiply numerators and denominators straight across.

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Dividing Fractions

Basic
ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Divide by multiplying by the reciprocal (flip and multiply).

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Average (Mean)

Basic
average=sum of valuesnumber of values\text{average} = \frac{\text{sum of values}}{\text{number of values}}

Add all values and divide by how many there are.

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Ratio

Basic
a:b=aba : b = \frac{a}{b}

Compares two quantities by division.

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Proportion (Cross Multiply)

Basic
ab=cd⇒ad=bc\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc

Two equal ratios; solve using cross multiplication.

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Unit Rate

Basic
unit rate=totalquantity\text{unit rate} = \frac{\text{total}}{\text{quantity}}

Cost or amount for a single unit.

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Speed = Distance ÷ Time

Basic
s=dts = \frac{d}{t}

How fast something moves.

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Distance = Speed × Time

Basic
d=s×td = s \times t

How far something travels.

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Rounding Rule

Basic
5 or more↑, 4 or less↓5\text{ or more} \uparrow, \ 4\text{ or less} \downarrow

Round up if the next digit is 5+, otherwise round down.

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Order of Operations (PEMDAS)

Basic
P→E→(M÷D)→(A÷S)P \to E \to (M \div D) \to (A \div S)

Parentheses, Exponents, Multiply/Divide, Add/Subtract.

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Place Value

Basic
N=∑di×10iN = \sum d_i \times 10^{i}

Each digit’s value depends on its position.

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Even and Odd

Basic
n mod 2=0⇒evenn \bmod 2 = 0 \Rightarrow \text{even}

A number is even if it divides by 2 with no remainder.

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Discount Price

Basic
price=N(1−p100)\text{price} = N \left(1 - \frac{p}{100}\right)

Final price after a percentage discount.

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Sales Tax Total

Basic
total=N(1+t100)\text{total} = N \left(1 + \frac{t}{100}\right)

Price after adding a percentage tax.

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Tip Amount

Basic
tip=p100×bill\text{tip} = \frac{p}{100} \times \text{bill}

Gratuity as a percentage of the bill.

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Percent Increase

Basic
new=old(1+p)new = old(1 + p)

Value after a percentage increase.

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Percent Decrease

Basic
new=old(1−p)new = old(1 - p)

Value after a percentage decrease.

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Mixed Number to Fraction

Basic
abc=ac+bca\frac{b}{c} = \frac{ac + b}{c}

Convert a mixed number to an improper fraction.

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Reciprocal

Basic
1x\frac{1}{x}

The multiplicative inverse of a number.

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Square of a Number

Basic
n2=n×nn^2 = n\times n

A number multiplied by itself.

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Cube of a Number

Basic
n3=n×n×nn^3 = n\times n\times n

A number multiplied by itself three times.

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Square Root

Basic
n\sqrt{n}

The value that squares to n.

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Divisibility by 3

Basic
digit sum divisible by 3\text{digit sum divisible by 3}

A number is divisible by 3 if its digit sum is.

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Divisibility by 9

Basic
digit sum divisible by 9\text{digit sum divisible by 9}

A number is divisible by 9 if its digit sum is.

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Percent to Fraction

Basic
p%=p100p\% = \frac{p}{100}

Convert a percentage to a fraction.

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