Consumer Arithmetic

Percentages

Per cent means "out of a hundred", so $15\%$ is $\dfrac{15}{100} = 0.15$.

• A percentage of a quantity — multiply. $15\%$ of $\$240$ is $0.15 \times 240 = \$36$.
• Increase by $r\%$ — multiply by $\left(1 + \dfrac{r}{100}\right)$.
• Decrease by $r\%$ — multiply by $\left(1 - \dfrac{r}{100}\right)$.
• Express one quantity as a percentage of another — divide, then multiply by 100.

Profit, loss and discount

The cost price goes on the bottom. This is the single most common error in the topic: dividing by the selling price instead gives a smaller, wrong answer.

A discount is a reduction on the marked price. A $15\%$ discount on $\$3200$ leaves $0.85 \times 3200 = \$2720$ — one multiplication, rather than working out the discount and subtracting it.

Ratio and proportion

To share an amount in the ratio $a : b$, add the parts, divide to find one part, then multiply.

For $\$450$ shared $4 : 5$: there are $4 + 5 = 9$ parts, so one part is $\$50$, giving $\$200$ and $\$250$.

Simple and compound interest

$P$ is the principal, $R$ the rate per cent per year, $T$ the time in years.

Simple interest is calculated on the original sum every year, so it is the same each year. Compound interest is calculated on the balance, so each year earns slightly more than the last.

$\$5000$ at $6\%$ compound for 3 years gives $5000 \times 1.06^3 = \$5955.08$, so the interest is $\$955.08$ — against $\$900$ if it had been simple.

Hire purchase

Buying by instalments costs more than paying cash. The total is

$$\text{deposit} + (\text{number of instalments} \times \text{instalment})$$

and the extra over the cash price is the cost of the credit.

Wages and salaries

• Wage — paid per hour or per day. Salary — a fixed annual amount, usually paid monthly.
• Overtime is paid at a higher rate, often "time-and-a-half" ($1.5\times$) or "double time" ($2\times$).

Overtime applies only to the hours beyond the basic week. For $\$18$ an hour with a 40-hour basic week and 46 hours worked:

$$40 \times \$18 + 6 \times \$27 = \$720 + \$162 = \$882$$