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5. Exponential Functions
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United States of AmericaFL
Grade 912

5.08 Compound interest

Lesson
Practice
Lesson

Concept summary

Compound interest is a method for computing interest where the interest is computed from the original principal combined with all interest accrued so far.

\displaystyle A=P\left(1+\frac{r}{n}\right)^{nt}
\bm{A}
future value, or final amount
\bm{P}
principal, or present value
\bm{r}
rate of interest per year
\bm{n}
number of times interest is compounded per year
\bm{t}
number of years

Because compound interest has the variable t in the exponent while P, r, and n are constant, it is considered an exponential function. We can see this in how a sum of money earning compound interest grows by a constant percent rate per unit of time.

We can think of compound interest as a repeated application of simple interest.

Worked examples

Example 1

Frasier's investment of \$ 200 earns interest at a rate of 4\% per year, compounded annually for 5 years.

a

Use the compound interest formula to find the final balance of his investment.

Approach

To apply the compound interest formula we need to identify which variables we have given information for. In this example we know that the principal is P=200, the rate is r=0.04 and the time is t=5. If the interest is compounded annually, then we also know n=1 since annually means once per year.

Solution

A=200\left(1+\dfrac{0.04}{1}\right)^{1\times5}=243.33.

Frasier will have \$243.33 after 5 years.

b

Frasier wanted to have \$300 at the end of his investment term. Find the initial amount would he need to invest at the same rate to have an ending value of \$300.

Approach

In this example we know the ending balance is A=300, the rate is still r=0.04, the time is still t=5 and the the interest is still compounded annually so n=1.

Solution

Substituting all given values into the formula gives us:

\displaystyle A\displaystyle =\displaystyle P\left(1+\frac{r}{n}\right)^{nt}Compound interest formula
\displaystyle 300\displaystyle =\displaystyle P\left(1+\frac{0.04}{1}\right)^{1\times 5}Substitute in values for the variables
\displaystyle 300\displaystyle =\displaystyle P(1.04)^{5}Simplify
\displaystyle 246.58\displaystyle =\displaystyle PDivide

Frasier would need to invest \$246.58 to have an ending balance of \$300 after investing his money for 5 years at a rate of 4\% compounded annually.

Example 2

A \$ 1610 investment earns interest at 4.5\% per year compounded quarterly over 10 years.

Use the compound interest formula to calculate the value of this investment to the nearest cent.

Solution

We are given that the present value is P=1610 and the annual rate is r=0.045. The investment is compounded quarterly which means four times per year so n=4 for t=10 years.

Substituting these values into the compound interest formula gives us:

\displaystyle A\displaystyle =\displaystyle P\left(1+\frac{r}{n}\right)^{nt}Compound interest formula
\displaystyle A\displaystyle =\displaystyle 1610\left(1+\frac{0.045}{4}\right)^{10\times 4}Substitute in values for the variables
\displaystyle A\displaystyle =\displaystyle 1610(1.01125)^{40}Simplify

If we evaluate the expression for A, rounding to two decimal places, we find that the value of this investment is \$2518.65

Example 3

Decide if each expression represents simple interest or compound interest and explain your decision.

a

A=500(1+0.02)^6

Approach

Simple interest will take the form, A=P(1+rt) and compound interest will take the form A=P\left(1+\dfrac{r}{n}\right)^{nt}.

Solution

Compound interest, since the expression is exponential.

Reflection

This is compound interest where P=500, r=0.02, n=1 and t=6.

b

A=500(1+0.05(3))

Approach

Simple interest will take the form, A=P(1+rt) and compound interest will take the form A=P\left(1+\dfrac{r}{n}\right)^{nt}.

Solution

Simple interest, since the expression is linear.

Reflection

This is simple interest where P=500, r=0.05 and t=3.

Outcomes

MA.912.AR.1.1

Identify and interpret parts of an equation or expression that represent a quantity in terms of a mathematical or real-world context, including viewing one or more of its parts as a single entity.

MA.912.AR.1.2

Rearrange equations or formulas to isolate a quantity of interest.

MA.912.AR.5.4

Write an exponential function to represent a relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.

MA.912.FL.3.2

Solve real-world problems involving simple, compound and continuously compounded interest.

MA.912.FL.3.4

Explain the relationship between simple interest and linear growth. Explain the relationship between compound interest and exponential growth and the relationship between continuously compounded interest and exponential growth.

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