Simple interest

It is one percent, counted t times

$Prt$ is not a new idea. $Pr$ is a percent of a number — one year’s interest — and $t$ counts how many years of it there are.

$$\underbrace{500 \times 0.04}_{\text{one year} \;=\; 20} \times \;3\; = 60$$

Seeing it that way makes the rearrangements follow rather than needing to be learned.

Rearranged

All three quantities are multiplied, so any one is found by dividing by the other two.

$$P = \frac{I}{rt} \qquad r = \frac{I}{Pt} \qquad t = \frac{I}{Pr}$$

The two errors worth guarding against

The rate must be a decimal, and the time must be in years

The rate. $r$ for $5\%$ is $0.05$. Using $5$ makes every answer a hundred times too big — large enough that the result is usually visibly absurd, such as an account earning more than it holds.

The time. Rates are quoted per year, so nine months is $\tfrac{9}{12} = 0.75$ years. Leaving it as $9$ charges nine years of interest on a nine-month loan.

Both mistakes multiply the answer by a large factor rather than nudging it, so both are caught by asking whether the size is plausible.

Interest is not the balance

$$\text{balance} = P + I$$

The formula returns the extra money only. $500 that earns $60 is worth $560, and a question asking what an account is worth needs that final addition.

Worked examples

$800 at 5% for 4 years.

$$I = 800 \times 0.05 \times 4 = \$160$$

$600 at 6% for 8 months.

$$t = \frac{8}{12} = \frac{2}{3} \quad\Rightarrow\quad I = 600 \times 0.06 \times \frac{2}{3} = \$24$$

$2000 earns $240 in 3 years — what rate?

$$r = \frac{240}{2000 \times 3} = 0.04 \quad\Rightarrow\quad 4\%$$

Simple against compound

Simple interest is charged on the original principal forever. Compound interest is charged on the current balance, so it earns interest on interest.

$1000 at $10\%$ reaches $2000 after ten years under simple interest, and $2594 under compound. They agree for the first year only, which is exactly why the difference is missed. See simple interest.