Exponential Decay Equations Practice Test

Advanced Algebra Practice Test: ACT math skills.

Exponential Decay Equations Practice Test

This test has 20 questions

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After-test decay field guide

Fading Signal Observatory

Exponential decay removes the same fraction of the current amount during each equal period. This guide helps you translate the story into a decay factor, choose between ordinary decay, half-life, and continuous models, and solve for any missing quantity.

Start: initial amountRemain: factor below oneRepeat: number of periods

Core patternEqual ratios over equal time intervals

Decay factorGreater than zero and less than one

Graph directionFalls quickly, then levels toward zero

Recognize the decay fingerprint

Decay is multiplicative. The amount lost usually becomes smaller because the same percentage is taken from a shrinking quantity.

The standard decay equation

The initial amount is multiplied by the same remaining factor once per period.

A (t) = A_{0} {(1 - r)}^{t}

Initial amount

A_{0}

is the value when time is zero.

Decay rate

r

is written as a positive decimal.

Remaining factor

1 - r

is multiplied repeatedly.

Select the correct decay model

The prompt usually supplies a percent loss, a half-life, or a continuous decay constant. Do not mix the parameters from these three forms.

Percent per period

Discrete decay

Use this form when the quantity loses a stated percent each hour, month, year, or other fixed interval.

A = A_{0} {(1 - r)}^{t}

A loss of

14 %

means that

86 %

remains, so the factor is

0.86

Half every interval

Half-life decay

Use the half-life as the time required for the amount to be multiplied by one-half.

A = A_{0} {\frac{1}{2}}^{\frac{t}{h}}

The exponent

\frac{t}{h}

counts how many half-life intervals have passed.

Continuous change

Continuous decay

Use base $e$ when the model states that decay occurs continuously.

A = A_{0} e^{- k t}

For decay, the exponent is negative when

k

is given as a positive decay constant.

Half-life as a repeated checkpoint

At each full half-life, exactly half of the preceding amount remains. Fractional half-lives are handled naturally by the exponent.

The amount never loses a fixed number

Each halving is based on the amount currently present. That is why the differences shrink while the ratios remain constant.

Start

A_{0}

remains.

One interval

\frac{A_{0}}{2}

remains.

Two intervals

\frac{A_{0}}{4}

remains.

Three intervals

\frac{A_{0}}{8}

remains.

A_{n} = A_{0} {\frac{1}{2}}^{n}

Worked observation cases

Keep the decay factor and exponent together until the final calculator step. Then interpret the result using the units and context in the prompt.

CASE 01

Calculate value after annual depreciation

An item is worth $1800$ dollars and loses $14 %$ of its value each year. Find its value after $5$ years.

Discrete

Observe

Initial value

Start with $1800$ .

Convert

Remaining factor

Subtract $0.14$ from one.

Count

Periods

Five years means five annual multiplications.

Evaluate

Round at the end

Money is normally rounded to the nearest cent.

V (5) = 1800 {(0.86)}^{5} \approx 846.77

Interpretation: about

846.77

dollars remains. Subtracting the same

252

dollars each year would describe linear depreciation, not percentage decay.

CASE 02

Use a fractional number of half-lives

A medicine starts at $240$ milligrams and has a half-life of $6$ hours. Find the amount after $15$ hours.

Half-life

Observe

Start

The initial amount is $240$ milligrams.

Divide

Count intervals

The elapsed time contains $\frac{15}{6}$ half-lives.

Model

Use one-half

The half-life factor is fixed.

Interpret

Keep units

The output is an amount in milligrams.

A (15) = 240 {\frac{1}{2}}^{\frac{15}{6}} \approx 42.43

Interpretation: approximately

42.43

milligrams remains. Do not round

\frac{15}{6}

down to two complete half-lives; decay continues during the additional three hours.

CASE 03

Evaluate continuous decay

A quantity begins at $500$ units and follows a continuous decay constant of $0.08$ per year. Find the amount after $9$ years.

Continuous

Observe

Model clue

The word continuous signals base $e$ .

Sign

Decay direction

Use a negative exponent with a positive decay constant.

Units

Match time

Both the constant and time use years.

Calculate

One final step

Evaluate the complete exponential expression.

A (9) = 500 e^{- (0.08) (9)} = 500 e^{- 0.72} \approx 243.38

Interpretation: about

243.38

units remains. A positive exponent would produce growth and contradict the stated behavior.

CASE 04

Find when a target fraction remains

A quantity retains $92 %$ of its value each year. How long until only $30 %$ of the initial amount remains?

Solve time

Ratio

Cancel the start

Use the fraction remaining on the left side.

Factor

Read carefully

The prompt already gives the remaining factor.

Logarithm

Release time

Take logarithms after isolating the power.

Meaning

Interpret years

The target is reached during the fifteenth year.

0.30 = {0.92}^{t}

t = \frac{\ln (0.30)}{\ln (0.92)} \approx 14.4

Interpretation: the continuous-time model reaches

30 %

after about

14.4

years. At whole annual checkpoints, first test year

15

Solve backward from observed data

When a rate or starting amount is missing, undo the model in reverse order. Keep the ratio of final amount to initial amount visible.

Recover the initial amount

Divide the observed amount by the complete decay power.

A_{0} = \frac{A (t)}{{(1 - r)}^{t}}

Build the remaining factor from the stated percent loss.

Raise the factor to the number of decay periods.

Divide the later amount by that result.

Recover the decay rate

If an amount falls from $900$ to $620$ in $3$ equal periods, first find the remaining factor.

b = {\frac{620}{900}}^{\frac{1}{3}} \approx 0.883

r = 1 - b \approx 0.117 = 11.7 %

The factor is the fraction retained; the decay rate is the fraction lost. They add to one.

Graph and table diagnostics

A decay curve stays positive, decreases, and approaches zero. A data table shows a constant ratio when time steps are equal.

What the shape must show

The curve drops steeply at first, then flattens. In a basic decay model with a positive initial value, it approaches the horizontal axis without becoming negative.

Intercept: the output at time zero equals the initial amount.

Ratio: divide consecutive outputs at equal time steps to find the decay factor.

Direction: a factor below one makes later values smaller.

Scale: the vertical decreases are not equal, even though the percent loss is equal.

\frac{A (t + 1)}{A (t)} = b

Model-selection reference

Match the wording to the correct structure, then check that every parameter uses compatible units.

Prompt clue	Best setup	Essential check
Loses the same percent per period	$A = A_{0} {(1 - r)}^{t}$	Convert the percent to a decimal before subtracting from one.
Retains a stated percent per period	Use the retained decimal directly as the factor.	Do not subtract the retained percent from one again.
Has a stated half-life	$A = A_{0} {\frac{1}{2}}^{\frac{t}{h}}$	Divide total time by the half-life in matching units.
Decays continuously	$A = A_{0} e^{- k t}$	The negative exponent makes the amount decrease.
Asks when a target remains	Isolate the exponential power and take logarithms.	Use the target-to-start ratio before solving for time.

Skills Covered

This topic connects percent reasoning, exponential models, logarithms, tables, graphs, and interpretation.

Recognize repeated percent loss as exponential decay.
Convert a decay rate into a remaining factor.
Use discrete, half-life, and continuous decay equations.
Evaluate an amount after whole or fractional periods.
Find an unknown initial amount, rate, or elapsed time.
Check whether a result is reasonable in context.

Common Mistakes

These traps often produce a neat calculation built from the wrong factor or time scale.

Using the decay rate itself as the remaining factor.
Subtracting the same fixed amount every period.
Confusing the percent lost with the percent retained.
Using years with a rate stated per month.
Rounding a fractional number of half-lives to a whole number.
Using a positive exponent in a continuous decay equation.

Final observation audit

Before selecting an answer, confirm the model, factor, time scale, direction, and interpretation.

Initial valueDoes time zero reproduce the stated starting amount?

FactorIs the multiplier between zero and one?

Time unitDoes the exponent count the correct periods?

DirectionIs the final amount smaller but still positive?

Answer formDid I report the requested amount, rate, or time?

Practice note: estimate before calculating. A remaining factor near one should produce gradual decay, while a much smaller factor produces a rapid drop. If the computed result grows or becomes negative, revisit the model before comparing answer choices.

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