Initial amount of an investment compounded continuously

You know what the account has to be worth at the end. Divide by e raised to rt instead of multiplying — and the answer must come out smaller.

01 Worked Example Video

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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 shorter horizon

How much should be invested now at 7% per year, compounded continuously, to have $2,000 in five years?

Round to the nearest cent.

A bigger target

How much should be invested now at 4.2% per year, compounded continuously, to have $1,200 in six years?

Round to the nearest cent.

Spot the direction

An account compounds continuously. A deposit today will be worth more or less than the target amount it grows into?

Type less or more.

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

$P = \dfrac{A}{e^{rt}}$ the same formula, solved for P

equivalently $P = Ae^{-rt}$ a negative exponent instead of a division

Both forms give the same number, so use whichever your calculator makes easier. Dividing is usually safer under pressure, because a mistyped minus sign in the exponent produces a wrong answer that still looks plausible.

Common slips

(1) Multiplied instead of dividing. That answers the other question on this exam. The starting amount has to be smaller than the target. (2) Dropped the minus sign in the Ae^−rt form. (3) Rounded e^rt before dividing. On a five-figure target that shifts the cents.

04 Practice — Step by Step

Same factor, other direction

How much should be invested now at an interest rate of 5.5% per year, compounded continuously, to have $7,500 in eight years?

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

Step 1

Which way?

You have the ending amount and want the starting one. Multiply or divide by e^rt?

Step 2

The growth factor

rt = 0.055 × 8. Compute e^rt, six decimal places.

Step 3

Divide

7,500 ÷ 1.5527072, to the nearest cent.

Q8 Q10

Initial amount of an investment compounded continuously

The check that costs five seconds