Finding a final amount in a word problem on exponential growth or decay

Same engine as compound interest — <em>A = P(1+r)<sup>t</sup></em> — but the “account” is now a population, a depreciating asset, or anything else that scales by a constant rate each period.

01 Worked Example Video

Every step of Finding a final amount in a word problem on exponential growth or decay explained on video. Pause anywhere; the embed scrolls independently of the page.

YOUTUBE

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Town population

A small town has a population of 8,000 people, growing at 3% per year.

What will the population be in 10 years? Round to the nearest whole person.

population =

Car depreciation

A car is worth $24,000. It loses 15% of its value each year.

What will the car be worth in 5 years? Round to the nearest dollar.

Decay → use (1 − r) = 0.85 inside the parentheses.

value = $

Bacterial culture

A bacterial culture starts with 500 cells and grows at 20% per hour.

How many cells after 6 hours? Round to the nearest whole cell.

cells =

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

A = P(1 + r/n)^nt P is principal, r is annual rate, n is periods/year, t is years

Annual compounding → n = 1 formula collapses to A = P(1 + r)^t

When n = 1 there’s no fraction inside the parenthesis. Compute (1 + r), raise to the t-th power (carry lots of decimals), then multiply by P. Round only at the final answer.

Common slips

(1) Rounded too early. Truncating (1.0115)² from 1.02313225 to 1.0231 costs about seven cents on a $2,100 principal. Carry the full decimal. (2) Used r = 1.15 instead of 0.0115. 1.15% shifts two places, not one. (3) Confused A with I. A is the full balance; I = A − P.

04 Practice — Step by Step

Practice this problem step by step

Invest $2,000 in a 5-year CD at 4% compounded annually. What’s the balance after 5 years?

Step 1

Convert the rate

Write 4% as a decimal.

r =

Step 2

Compute (1 + r) to the t-th power

Compute (1.04)⁵ and carry at least 5 decimal places.

(1.04)⁵ ≈

Step 3

Multiply by P

Compute 2,000 × 1.2166529. Round to the nearest cent.

A = $

Q17 Q19

Finding a final amount in a word problem on exponential growth or decay

You walked the compound formula end to end.