# Lesson 17

Different Compounding Intervals

Let's find out what happens when we repeatedly apply the same percent increase at different intervals of time.

### Problem 1

The population of a city in 2010 is 50,000, and it grows by 5% each year after.

- Write a function \(f\) which models the population of the city \(t\) years after 2010.
- What is the population of the city in 2017?
- What will the population of the city be in 2020? What about in 2030?
- By what factor does the population grow between 2010 and 2020? What about between 2020 and 2030?

### Problem 2

A person charges $100 to a credit card with a 24% nominal annual interest rate.

Assuming no other charges or payments are made, find the balance on the card, in dollars, after 1 year if interest is calculated:

- annually
- every 6 months
- every 3 months
- monthly
- daily

### Problem 3

A couple has $5,000 to invest and has to choose between three investment options.

- Option A: \(2\frac{1}{4} \%\) interest applied each quarter
- Option B: \(3\%\) interest applied every 4 months
- Option C: \(4\frac{1}{2}\%\) interest applied twice each year

If they plan on no deposits and no withdrawals for 5 years, which option will give them the largest balance after 5 years? Use a mathematical model for each option to explain your choice.

### Problem 4

Elena says that 6% interest applied semi-annually is the same as 1% interest applied every month: she reasons they are the same because they are both a 12% nominal annual interest rate.

- Is Elena correct that these two situations both offer a 12% nominal annual interest rate?
- Is Elena correct that the two situations pay the same amount of interest?

### Problem 5

A bank pays 8% annual interest, compounded at the end of each month. An account starts with $600, and no further withdrawals or deposits are made.

- What is the monthly interest rate?
- Write an expression for the account balance, in dollars, after one year.
- What is the effective annual interest rate?
- Write an expression for the account balance, in dollars, after \(t\) years.

### Problem 6

At the end of each year, 10% interest is charged on a $500 loan. The interest applies to any unpaid balance on the loan, including previous interest.

Select **all** the expressions that represent the loan balance after two years if no payments are made.

\(500 + 2 \boldcdot (0.1) \boldcdot 500\)

\(500 \boldcdot (1.1) \boldcdot (1.1)\)

\(500 + (0.1) + (0.1)\)

\(500 \boldcdot (1.1)^2\)

\((500 + 50) \boldcdot (1.1)\)

### Problem 7

Here is a graph of the function \(f\) given by \(f(x) = 100 \boldcdot 2^x\).

Suppose \(g\) is the function given by \(g(x) = 50 \boldcdot (1.5)^x\).

Will the graph of \(g\) meet the graph of \(f\) for any positive value of \(x\)? Explain how you know.

### Problem 8

Suppose \(m\) and \(c\) each represent the position number of a letter in the alphabet, but \(m\) represents the letters in the original message, and \(c\) represents the letters in a secret code.

The equation \(c = m + 7\) is used to encode a message.

- Write an equation that can be used to decode the secret code into the original message.
- What does this code say: "AOPZ PZ AYPJRF!"?