About this video
The video solves the compound interest formula A = P(1 + r)^t for the interest rate r. Dividing by the principal, raising both sides to the power 1/t, and subtracting one isolates r, giving r = (A/P)^(1/t) - 1. The key idea is that a power can be undone with the reciprocal power, and that exponent rules make the algebra work.
You'll be able to
- rearrange the compound interest formula to solve for the rate r.
- use exponent rules to undo a power.
- check a solved formula by substituting values back into the original.
Key words
- Principal (P)
- The initial amount invested or borrowed.
- Future value (A)
- The amount an investment is worth after t years.
- Compounded annually
- Interest is added to the balance once each year, and then earns interest itself.
- Exponent rule
- Raising a power to another power multiplies the exponents together.
- Inverse operation
- An operation that undoes another, such as raising to the 1/t power undoing a power of t.
Transcript
- [Instructor] If an initial investment, P, bears interest at a rate r, so this initial investment P, bears interest at a rate R, and is compounded annually, its future value A, after t years, can be determined with the equation above. Which of the following equations shows the interest rate in terms of the future value, initial investment, and number of years invested?
So they really just want us to solve for r. So let's see if we can do that. So we have the future value, gonna give myself some space. Future value is equal, I'm just rewriting it, is equal to the initial investment times one plus our interest rate, and then that quantity to the t power.
So let's see what we can do. So the first thing I could do is I can get the P onto the left-hand side by dividing both sides of this equation by our initial investment. So if I do that, I get, and actually, let me, let me swap the sides, too. Then I get one plus r to the t power is equal to our future value divided by our initial investment.
Now, how do I get rid of this to the t power here? Well, I can raise both sides to the one over t power. So I could raise that to the one over t power. One over t. By doing the left side, I have to do it to the right side, as well. So on the left-hand side, if I raise something to the t and then I raise it to the one over t, remember, if you raise something to an exponent and then raise it to another exponent, you're raising it to the product of these two exponents.
So this is equivalent to raising one, this is equivalent to just raising one plus r to the first power, or this is just going to simplify to one plus r on the left-hand side. And on the right-hand side, we're gonna have our future value divided by our initial investment to the one over t power.
And now, pretty straightforward. You wanna solve for r, subtract one from both sides. We get r is equal to our future value divided by our initial investment to the one over t power minus one. And that is this choice right over there.
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