Arithmetic & Basics

Everyday number sense: fractions, percentages, ratios and averages

Arithmetic & Basics brings together the everyday number skills you use all the time, from percentages and fractions to ratios, averages, and unit rates. It’s ideal for students, exam prep, and anyone handling real-world calculations at work or in daily life, because these formulas help you compare, measure, convert, and solve problems quickly and accurately.

Percentage

\% = \frac{\text{part}}{\text{whole}} \times 100

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

Percent of a Number

\text{value} = \frac{p}{100} \times N

Find p percent of a number N.

Percentage Change

\% \text{change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100

Increase or decrease of a value as a percentage.

Fraction to Decimal

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

Convert a fraction into a decimal by dividing.

Adding Fractions

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

Add two fractions using a common denominator.

Multiplying Fractions

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

Multiply numerators and denominators straight across.

Dividing Fractions

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

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

Average (Mean)

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

Add all values and divide by how many there are.

Ratio

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

Compares two quantities by division.

Proportion (Cross Multiply)

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

Two equal ratios; solve using cross multiplication.

Unit Rate

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

Cost or amount for a single unit.

Speed = Distance ÷ Time

s = \frac{d}{t}

How fast something moves.

Distance = Speed × Time

d = s \times t

How far something travels.

Rounding Rule

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

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

Order of Operations (PEMDAS)

P \to E \to (M \div D) \to (A \div S)

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

Place Value

N = \sum d_i \times 10^{i}

Each digit’s value depends on its position.

Even and Odd

n \bmod 2 = 0 \Rightarrow \text{even}

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

Discount Price

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

Final price after a percentage discount.

Sales Tax Total

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

Price after adding a percentage tax.

Tip Amount

\text{tip} = \frac{p}{100} \times \text{bill}

Gratuity as a percentage of the bill.

Percent Increase

new = old(1 + p)

Value after a percentage increase.

Percent Decrease

new = old(1 - p)

Value after a percentage decrease.

Mixed Number to Fraction

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

Convert a mixed number to an improper fraction.

Reciprocal

\frac{1}{x}

The multiplicative inverse of a number.

Square of a Number

n^2 = n\times n

A number multiplied by itself.

Cube of a Number

n^3 = n\times n\times n

A number multiplied by itself three times.

Square Root

\sqrt{n}

The value that squares to n.

Divisibility by 3

\text{digit sum divisible by 3}

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

Divisibility by 9

\text{digit sum divisible by 9}

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

Percent to Fraction

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

Convert a percentage to a fraction.