Interest is a form of payment from you to your bank, or from the bank to you, based on a principal balance. For example, when you put money in a savings account, you receive payment from the bank for letting them borrow your money. Likewise, when you take out a loan from the bank, you pay money to the bank in addition to the amount you borrowed. This is called interest. Interest is a percentage, and therefore one way to calculate interest is by using the growth factor.

The formula below looks very similar to the one for growth factor, and it is. You can think of ${K}_{0}$ as an old value, and ${K}_{n}$ as a new value. The growth factor is the same, but there is also an exponent, which tells you how many time units the calculation covers. Here you’ll only look at one time unit.

Formula

$${K}_{n}={K}_{0}\phantom{\rule{-0.17em}{0ex}}{\left(1+\frac{p}{100}\right)}^{n}$$ |

where ${K}_{0}$ is the initial balance, and $p$ is the interest rate. After $n$ periods (or number of times the interest is added) the new balance is ${K}_{n}$.

Example 1

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You put $\text{\$}\text{}10\phantom{\rule{0.17em}{0ex}}000\text{}$ in a savings account and receive a fixed interest of $\text{}4\text{}\phantom{\rule{0.17em}{0ex}}\text{\%}$ per year. How much money do you have in the bank after 1 year?

$$\begin{array}{llll}\hfill {K}_{1}& =10\phantom{\rule{0.17em}{0ex}}000\cdot \phantom{\rule{-0.17em}{0ex}}{\left(1+\frac{4}{100}\right)}^{1}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =10\phantom{\rule{0.17em}{0ex}}000\cdot 1.04\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =10\phantom{\rule{0.17em}{0ex}}400\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$ After 1 year you have $$10\phantom{\rule{0.17em}{0ex}}400$ in the bank.

Example 2

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You’re going to borrow $\text{\$}\text{}5000\text{}$ from the bank to finance your graduation party. The bank thinks this is a bad use of the money, and charges an interest rate of $\text{}40\text{}\phantom{\rule{0.17em}{0ex}}\text{\%}$. How much do you have to pay back to the bank after 1 year?

$$\begin{array}{llll}\hfill {K}_{1}& =5000\cdot \phantom{\rule{-0.17em}{0ex}}{\left(1+\frac{40}{100}\right)}^{1}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =5000\cdot 1.40\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =7000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$ After 1 year you have to pay $$7000$ back to the bank. Notice that the bank is concerned you won’t be able to pay the loan back, so they are charging you

$$\text{\$}7000-\text{\$}5000=\text{\$}2000$$ |

in interest!

Example 3

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Last year you put $\text{\$}\text{}2000\text{}$ into a savings account. This year you have $\text{\$}\text{}2100\text{}$ in the account. What’s the interest on the savings account?

The starting balance is ${K}_{0}=2000$, the final balance is ${K}_{1}=2100$, and the number of time periods is $n=1$ as only one year has passed. Now you just have to insert these figures right into the formula, and solve for $p$:

$$\begin{array}{llllllll}\hfill {K}_{1}& ={K}_{0}\phantom{\rule{-0.17em}{0ex}}{\left(1+\frac{p}{100}\right)}^{1}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 2100& =2000\phantom{\rule{-0.17em}{0ex}}\left(1+\frac{p}{100}\right)\phantom{\rule{2em}{0ex}}& \hfill & |\xf72000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 1.05& =\phantom{\rule{-0.17em}{0ex}}\left(1+\frac{p}{100}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 0.05& =\frac{p}{100}\phantom{\rule{2em}{0ex}}& \hfill & |\cdot 100\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 5& =p\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill p& =5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$The interest on the savings account is therefore $5$ %.

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