Exponential Equation Word Problems Practice Test

Advanced Algebra Practice Test: ACT math skills.

Exponential Equation Word Problems Practice Test

This test has 20 questions

Instant feedback · Worked explanations

After-test scenario field guide

Exponential Scenario Signal Atlas

Word problems hide exponential structure inside a story. Your job is to find the starting amount, decide whether a quantity grows or decays, translate the rate into a factor, match the time unit, and interpret the requested output.

Clue repeated percentModel exponentialUnknown locate itCheck interpret it

1 ReadWhat changes?

2 ClassifyGrowth or decay?

3 BuildInitial value and factor

4 SolveForward or backward

5 ReportValue with units

Decode the story before calculating

Different phrases point to different exponential structures. Circle the time unit, underline the rate or interval, and identify the amount at time zero.

Growth

Increases by a percent

Add the decimal rate to one.

b = 1 + r

Decay

Decreases by a percent

Subtract the decimal rate from one.

b = 1 - r

Doubling

Doubles every interval

Use base two and count doubling intervals.

Q = Q_{0} 2^{\frac{t}{d}}

Half-life

Halves every interval

Use one-half and count half-life intervals.

Q = Q_{0} {\frac{1}{2}}^{\frac{t}{h}}

Finance

Compounded each year

Use the periodic rate and total number of periods.

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

Choose a route through the problem

The same model can be used forward to find a future amount or backward to recover time, rate, or an initial value.

Future amount

All model parameters are known. Substitute and evaluate the exponential expression.

Forward route

Elapsed time

The variable is in the exponent. Isolate the power and use logarithms.

Logarithm route

Unknown rate

Divide by the start, take the appropriate root, then convert factor to rate.

Root route

Initial amount

Divide the observed amount by the complete growth or decay power.

Division route

Worked scenario missions

Each solution separates translation from calculation. That prevents story details from being placed in the wrong part of the equation.

MISSION 1

Doubling population

A culture begins with $150$ organisms and doubles every $3$ hours. Estimate the amount after $10$ hours.

Growth

Start

Initial amount

Use $150$ .

Factor

Doubling base

Each interval multiplies by $2$ .

Intervals

Divide time

The exponent is $\frac{10}{3}$ .

Report

Count context

Round to a whole organism.

Q (10) = 150 2^{\frac{10}{3}} \approx 1512

Interpretation: the model predicts about

1512

organisms. Ten hours contains more than three doubling intervals, so the answer must exceed

150 \cdot 2^{3}

MISSION 2

Vehicle depreciation

A vehicle is worth $28000$ dollars and loses $16 %$ of its value each year. Find its value after $5$ years.

Decay

Start

Initial value

Use $28000$ .

Remain

Decay factor

Subtract $0.16$ from one.

Time

Annual periods

Use an exponent of $5$ .

Report

Money format

Round to the nearest cent.

V (5) = 28000 {(0.84)}^{5} \approx 11709.93

Interpretation: about

11709.93

dollars remains. The factor is

0.84

, not

0.16

, because the model multiplies by the fraction retained.

MISSION 3

Quarterly compound interest

Invest $4200$ dollars at $5.1 %$ annual interest compounded quarterly for $4$ years.

Finance

Principal

Starting balance

Use $4200$ .

Rate

Decimal annual rate

Use $0.051$ .

Clock

Quarterly

Use $n = 4$ .

Periods

Total count

There are $4 \cdot 4 = 16$ periods.

A = 4200 {(1 + \frac{0.051}{4})}^{16} \approx 5143.81

Interpretation: the balance is about

5143.81

dollars, so the interest earned is

943.81

dollars.

MISSION 4

Half-life target time

A medicine begins at $200$ milligrams and has a half-life of $8$ hours. When will $25$ milligrams remain?

Find time

Ratio

Target over start

Use $\frac{25}{200}$ .

Pattern

Recognize a power

The ratio equals $\frac{1}{8}$ .

Intervals

Count halvings

Three halvings produce one-eighth.

Time

Multiply

Use three intervals of $8$ hours.

\frac{25}{200} = {\frac{1}{2}}^{\frac{t}{8}} = {\frac{1}{2}}^{3}

\frac{t}{8} = 3 t = 24

Interpretation:

25

milligrams remains after

24

hours. No logarithm is necessary because the target ratio is an exact power of one-half.

MISSION 5

Recover an annual growth rate

A town grows from $18000$ people to $23500$ people in $6$ years. Assume a constant annual percent rate.

Find rate

Substitute

Known values

Place the start, end, and time in the growth model.

Divide

Isolate power

Divide by $18000$ .

Root

Undo six years

Take the sixth root.

Convert

Factor to rate

Subtract one and write a percent.

r = {\frac{23500}{18000}}^{\frac{1}{6}} - 1 \approx 0.0454

Interpretation: the annual growth rate is approximately

4.54 %

. Dividing the total percent increase by six ignores compounding.

Locate the unknown before choosing an operation

Do not automatically use logarithms. They are needed when the variable remains in an exponent after the exponential expression is isolated.

Four common unknowns

Start from the general model $Q = Q_{0} b^{t}$ , then work backward in the correct order.

Final amount

Substitute all known values and evaluate.

Initial amount

Divide by the complete exponential factor.

Growth or decay factor

Divide first, then take the root determined by time.

Elapsed time

Divide first, then use logarithms.

Q_{0} = \frac{Q}{b^{t}}

Run a reasonableness radar

A quick estimate can reveal a wrong sign, reversed factor, lost time conversion, or answer that does not match the story.

Four questions before selecting an answer

An exponential result should agree with both the equation and the real meaning of the situation.

Direction: should the amount increase or decrease?

Scale: does the number of doubling, halving, or percent-change periods support the size?

Time: did the exponent count the correct units?

Output: did the question ask for an amount, rate, time, or change?

Scenario translation reference

Match the clue to its mathematical meaning, then perform the final control check.

Story clue	Translation	Control check
Increases by the same percent	$b = 1 + r$	The factor is greater than one.
Decreases by the same percent	$b = 1 - r$	The factor is between zero and one.
Doubles or halves every interval	Divide total time by the stated interval.	Fractional intervals remain in the exponent.
Compounded monthly or quarterly	Divide the annual rate and multiply the years by the frequency.	The frequency appears in two places.
Asks when a target is reached	Isolate the exponential power and use logarithms.	Check whether whole periods are required.

Skills Covered

This topic combines reading comprehension, percent reasoning, exponential equations, logarithms, and interpretation.

Recognize growth, decay, doubling, half-life, and compound-interest situations.
Translate a verbal rate into a growth or decay factor.
Match the exponent to the correct time interval.
Calculate future amounts and remaining quantities.
Find an unknown time, rate, factor, or initial amount.
Explain a numerical result using units and context.

Common Mistakes

These errors come from translating the story incorrectly even when the later arithmetic is accurate.

Using the percent rate itself as the exponential factor.
Treating equal percent change as equal additive change.
Confusing the amount lost with the amount remaining.
Mixing hourly, monthly, and yearly time units.
Using logarithms before isolating the exponential power.
Reporting an intermediate factor instead of the requested quantity.

Final scenario audit

Use these five checkpoints to verify the translation, calculation, and meaning of your answer.

StartDid I identify the amount at time zero?

DirectionDoes the factor produce growth or decay as stated?

ClockDoes the exponent count the correct periods?

UnknownDid I solve for the quantity actually requested?

MeaningDid I round appropriately and include units?

Practice note: write one sentence describing what the factor means before using a calculator. A factor is the multiplier applied during one period. This habit separates the rate from the remaining fraction and makes the exponent easier to interpret.

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