**Basic Mathematical Concepts in Business**

**Ratios**

A **ratio** is a comparison of two numbers by division. It helps to determine the relationship between two quantities.

The ratio of $a$ to $b$ can be expressed as:

$a : b quad text{or} quad frac{a}{b}$

**Rates**

A **rate**, like a ratio, is a comparison of two quantities, but the quantities may have different units of measure. For example, speed is measured in km/hr, comparing distance and time.

A rate that has a denominator of 1 is called a **unit rate**.

The **rate of change** of a given quantity is used to measure how one quantity changes in relation to another:

$text{Rate of Change} = frac{text{Final Amount – Original Amount}}{text{Original Amount}}$

**Proportions**

A **proportion** is an equation that states that two ratios are equal.

In the proportion $a : b = c : d$, with $b neq 0$ and $d neq 0$, the four numbers are referred to as the terms of the proportion.

The first and last terms ($a$ and $d$) are called the **extremes**, and the second and third terms ($b$ and $c$) are called the **means**.

For three quantities $a$, $b$, and $c$, if:

$frac{a}{b} = frac{b}{c} quad Rightarrow quad b^2 = a times c$

This is called the **mean proportion** between $a$ and $c$.

**Compound Proportions**

A **compound proportion** is when one variable quantity depends on two or more other variable quantities.

Two quantities $x$ and $y$ are said to be in **inverse proportion** if an increase in $x$ causes a proportional decrease in $y$:

$x times y = k, quad text{where } k text{ is a constant}$

If $x$ and $y$ vary directly (in **direct proportion**), then:

$y = kx, quad text{where } k text{ is a constant}$

**Percentages**

A **percentage** is a number or ratio expressed as a fraction of 100. It is used widely in business to describe profit margins, discounts, and markups.

The difference between a product’s selling price and its cost is called **markup**:

$text{Markup} = text{Selling Price} – text{Purchase Price}$