Final amount of a loan or investment compounded continuously

The one compounding formula with no n in it. Multiply the principal by e raised to rt — one calculator keystroke sequence, and the whole question.

01 Worked Example Video

▸ VIDEO COMING SOON

A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

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.

A borrowed amount

$1,700 is borrowed for three years at 5% per year, compounded continuously.

Find the amount owed. Round to the nearest cent.

A longer term

$5,000 is invested for ten years at 3.1% per year, compounded continuously.

What is it worth at the end? Round to the nearest cent.

Just the exponent

An account pays 4.25% compounded continuously and is held for six years.

What is rt? Give the decimal.

rt =

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

$A = Pe^{rt}$ no n: nothing is compounded a set number of times

r as a decimal, t in years 5% for 3 years means rt = 0.15

Every other compounding formula divides the rate by n and multiplies the years by n. Continuous compounding is the limit of doing that forever, so the n disappears and what is left is e to the power of rate times time.

Common slips

(1) Used the percent instead of the decimal. e&sup5; is astronomically wrong; e^0.05t is right. (2) Looked for an n. There is not one. If the question says continuously, quarterly and monthly are irrelevant. (3) Rounded e^rt early. ALEKS says do not round intermediate computations, and on large principals a fourth decimal place moves the cents.

04 Practice — Step by Step

Rate times time, then e

Suppose $2,400 is borrowed for four years at an interest rate of 6% per year, compounded continuously.

Find the amount owed, assuming no payments are made until the end.

Do not round any intermediate computations, and round your answer to the nearest cent.

Step 1

The exponent

Work out rt: the rate as a decimal, times the years.

rt =

Step 2

Raise e to it

Compute e^0.24. Carry at least six decimal places.

Step 3

Multiply by the principal

2,400 × 1.2712492, to the nearest cent.

Q7 Q9

Final amount of a loan or investment compounded continuously

How much continuous actually buys you