Mathematics and Statistics · Ch 18 — Commercial Mathematics

Percentage and its Commercial Use

This Maharashtra Std XI (Commerce) Mathematics and Statistics chapter gathers the everyday arithmetic of trade — percentages, profit and loss, discount, commission and brokerage, interest, shares and GST — into one toolkit. Every idea rests on the single notion of a percentage, so we begin there. The chapter draws on the same standard, well-established treatment of commercial arithmetic used in mathematics curricula nationally.

What a percentage means

Percentage

A percentage is a fraction whose denominator is $100$ . " $r$ percent", written $r\%$ , means $\dfrac{r}{100}$ . Thus $r\%$ of a quantity $Q$ is

$r\%\ \text{of}\ Q=\frac{r}{100}\times Q.$

A percentage is just a convenient way of comparing to a base of $100$ . To turn a fraction or decimal into a percentage, multiply by $100$ ; to turn a percentage back into a decimal, divide by $100$ . For example $\dfrac{3}{4}=0.75=75\%$ , and $12\%=0.12$ .

Percentage increase and decrease

In business we constantly ask by what percentage a quantity has changed. The change is always measured against the original (base) value.

Note

Percentage change

$\text{percentage change}=\frac{\text{change in value}}{\text{original value}}\times100\%.$

A positive result is an increase, a negative result a decrease. The base is always the original amount, never the new one.

Tip

Multiplying factors are faster than "find the change, then add"

Increasing a quantity by $r\%$ multiplies it by $\left(1+\dfrac{r}{100}\right)$ ; decreasing it by $r\%$ multiplies it by $\left(1-\dfrac{r}{100}\right)$ . So a $15\%$ rise on $\text{₹}80{,}000$ is $80000\times1.15=\text{₹}92{,}000$ in one step — the same multiplying-factor idea powers the profit, discount and interest formulae that follow.

Definition 1Percentage

A fraction with denominator $100$ ; $r\%=\dfrac{r}{100}$ , so $r\%$ of $Q$ is $\dfrac{r}{100}\times Q$ . Convert a fraction/decimal to a percentage by multiplying by $100$ .

Definition 2Percentage change

$\dfrac{\text{change}}{\text{original value}}\times100\%$ , measured against the original (base) amount. Positive = increase, negative = decrease.

Definition 3Multiplying factor

Increasing by $r\%$ multiplies by $\left(1+\tfrac{r}{100}\right)$ ; decreasing by $r\%$ multiplies by $\left(1-\tfrac{r}{100}\right)$ — a one-step way to apply any percentage change.

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