Compound Interest Exponential Equations Practice Test

Advanced Algebra Practice Test: ACT math skills.

Compound Interest Exponential Equations Practice Test

This test has 20 questions

Instant feedback · Worked explanations

After-test interest field guide

Compound Interest Clockwork Bank

Compound interest adds earned interest to the balance, so later interest is calculated on a growing amount. This review shows how principal, rate, compounding frequency, and time fit into one exponential model.

DepositIdentify the principal.

ClockMatch the compounding periods.

BalanceSeparate total amount from interest earned.

PrincipalStarting account balance

Annual ratePercent written as a decimal

FrequencyCompounding periods per year

TimeUsually measured in years

Read the account formula like a ticket

Every symbol has a specific financial meaning. Label the quantities before substituting, especially the two appearances of the compounding frequency.

Periodic compound interest

The periodic rate is the annual rate divided by the number of compoundings per year. The exponent counts the total number of compounding periods.

A = P {(1 + \frac{r}{n})}^{n t}

Final amount

A

is principal plus earned interest.

Principal

P

is the initial deposit.

Annual rate

r

is the decimal annual rate.

Frequency

n

is periods per year, while

t

is years.

Set the compounding clock correctly

The frequency changes both the periodic rate and the total number of periods. These two changes must happen together.

Annually

One compounding period per year.

n = 1

Quarterly

Four compounding periods per year.

n = 4

Monthly

Twelve compounding periods per year.

n = 12

Daily

Many school problems use $365$ periods per year unless another convention is stated.

n = 365

Periodic or continuous compounding?

Use the wording of the problem. A stated schedule uses the periodic formula; the word continuously signals a different model with base $e$ .

Periodic compounding

Interest is added at specific checkpoints such as every month or every quarter.

A = P {(1 + \frac{r}{n})}^{n t}

Divide the annual rate by $n$ and multiply years by $n$ .

Continuous compounding

Interest is modeled as being added at every instant.

A = P e^{r t}

Keep the annual decimal rate in the exponent; do not add a frequency.

Worked transaction slips

Each example follows the same routine: list the variables, build the periodic rate and period count, evaluate once, and answer the exact financial question.

SLIP 01

Monthly compounding

Deposit $2500$ dollars at $4.8 %$ annual interest, compounded monthly for $3$ years.

Find amount

Principal

Starting balance

Use $P = 2500$ .

Rate

Decimal form

Use $r = 0.048$ .

Frequency

Monthly clock

Use $n = 12$ .

Time

Three years

The exponent contains $12 \cdot 3$ .

A = 2500 {(1 + \frac{0.048}{12})}^{12 \cdot 3} \approx 2886.38

Final balance:

2886.38

dollars. The interest earned is

2886.38 - 2500 = 386.38

dollars.

SLIP 02

Quarterly compounding

Invest $6000$ dollars at $5.2 %$ annual interest, compounded quarterly for $6$ years.

Find amount

Convert

Annual rate

Write the percent as $0.052$ .

Divide

Periodic rate

Use $\frac{0.052}{4}$ .

Count

Total periods

There are $4 \cdot 6 = 24$ quarters.

Evaluate

Round last

Keep all calculator digits until the final cent.

A = 6000 {(1 + \frac{0.052}{4})}^{4 \cdot 6} \approx 8180.46

Final balance: approximately

8180.46

dollars. The exponent is

24

, not

6

, because compounding occurs four times per year.

SLIP 03

Continuous compounding

Deposit $3200$ dollars at $3.9 %$ annual interest, compounded continuously for $7$ years.

Continuous

Signal

Model word

The word continuously selects base $e$ .

Principal

Starting balance

Use $P = 3200$ .

Exponent

Rate times years

Multiply $0.039$ by $7$ .

Answer

Money format

Round the final balance to two decimal places.

A = 3200 e^{(0.039) (7)} \approx 4204.48

Final balance: approximately

4204.48

dollars. There is no frequency value to substitute in the continuous model.

SLIP 04

Solve for investment time

How long does $1800$ dollars take to reach $2500$ dollars at $4.5 %$ annual interest compounded monthly?

Use logarithms

Substitute

Known values

Use the start, target, annual rate, and monthly frequency.

Divide

Isolate power

Divide both sides by the principal.

Logarithm

Release exponent

Apply a logarithm to both sides.

Solve

Divide carefully

The factor $12$ remains outside the logarithm.

\frac{2500}{1800} = {(1 + \frac{0.045}{12})}^{12 t}

t = \frac{\ln (2500 / 1800)}{12 \ln (1 + 0.045 / 12)} \approx 7.31

Time: about

7.31

years. If the account is checked only after complete monthly periods, test the next whole month before reporting when the target is first met.

Reverse-engineer the account

When the unknown is not the final balance, isolate the exponential structure first. Then choose division, roots, or logarithms according to where the unknown appears.

Unknown principal, rate, or time

The location of the variable determines the inverse operation.

Principal unknown: divide the final balance by the complete compound-growth power.

Rate unknown: divide by principal, take the root determined by total periods, subtract one, then multiply by the frequency.

Time unknown: divide by principal and use logarithms to bring the exponent down.

Final check: substitute the result into the original account equation.

P = \frac{A}{{(1 + r / n)}^{n t}}

Compare rates with effective annual yield

The quoted annual rate does not show the full effect of compounding. The effective annual rate measures the actual one-year percent increase.

Same quoted rate, slightly different growth

More frequent compounding produces a slightly larger balance when principal, quoted annual rate, and time are the same.

EAR = {(1 + \frac{r}{n})}^{n} - 1

For a nominal annual rate of $4.8 %$ compounded monthly, the effective annual rate is approximately $4.91 %$ .

Quick model-selection table

Use the prompt clue to select the equation and the final column to catch the most common setup error.

Prompt clue	Required action	Control check
Compounded monthly or quarterly	Use $A = P {(1 + r / n)}^{n t}$ .	The same $n$ divides the rate and multiplies time.
Compounded continuously	Use $A = P e^{r t}$ .	Do not insert a compounding frequency.
Interest earned	Calculate the balance, then use $I = A - P$ .	Do not report the total balance as interest.
Time to reach a target	Divide by principal, take logarithms, and solve for time.	Return time in the unit requested.
Effective annual rate	Calculate the one-year growth factor, then subtract one.	Convert the final decimal to a percent.

Skills Covered

This topic combines percent conversion, exponent rules, logarithms, financial vocabulary, and careful interpretation.

Identify principal, balance, interest, annual rate, frequency, and time.
Build periodic and continuous compound-interest models.
Convert a nominal annual rate to a periodic rate.
Count the total number of compounding periods.
Solve for principal, rate, or time using inverse operations.
Calculate interest earned and effective annual yield.

Common Mistakes

Most incorrect answers come from a unit mismatch or from confusing the account balance with the interest earned.

Entering a percent without converting it to a decimal.
Dividing the rate by frequency but forgetting to multiply the exponent.
Using months as years or years as months.
Reporting the final balance when only interest was requested.
Using the periodic formula for continuous compounding.
Rounding the periodic rate before evaluating the power.

Final account audit

Run these five checks before choosing an answer. They verify the model, units, and financial meaning.

PrincipalDid I identify the starting deposit rather than the target?

RateIs the annual percent written as a decimal?

ClockDoes the compounding frequency appear in both required places?

TimeAre the time units compatible with the annual rate?

RequestDoes the answer give balance, interest, rate, or time as asked?

Practice note: list

P

r

n

, and

t

before substitution. Estimate the balance first, then use that estimate to check the calculator result.

The examples in this review block are illustrative and are not copies of the test questions.