 The equation of value has five types of variables. The final balance B, the deposit or withdrawal amount ctk, the times of deposit withdrawal tk, the accumulation function a of t, and the investment period tau, given any four we can solve for the fifth. So, suppose Professor Jeff, still hoping to retire, is able to deposit $1,000 now and $1,000 next year. If he wants to be able to withdraw $500,000 in two years, what interest rate will he need to receive? So, we can write our equation of value. It will be convenient to assume the account is closed by a $500,000 withdrawal at the end, so our final balance is zero. We have deposits of $1,000 at t equals zero and t equals one, and we also have a withdrawal of $500,000 at t equals two. Assuming we're getting compound interest, then a of t is one plus i to the t, and so we need a of zero, a of one, and a of two, which are, and we have everything except for a of tau, and so we need to know what's tau. Well, we'll employ the time-honored tactic of procrastination. Note we can remove a factor of a of tau from our equation, and we multiply through by one plus i squared we get. To solve this equation, we'll let x equal one plus i, and reduce it to quadratic. Now, we note every coefficient has a factor of 1,000, so we can reduce our equation to, which has solutions, and so Professor Jeff needs to find a fund giving 2,087% interest each year. So, for example, let's say you borrow $1,000 at 10% annual interest rate, you pay back two installments of $600 equally spaced. When should the payments be made? So, we'll write down our equation of value, and we want our balance to be zero, so we can replace that, and tau is, well, we don't really know that, so we'll use a time-honored strategy and procrastinate. Now, we do know that we receive $1,000 at t equals zero. Since this is a withdrawal, then it's going to be a negative about, and we have payments of $600 at some time t, which is what we're looking at. The other thing to realize is that we get a second payment at 2t, because our payments are equally spaced, and at that point the loan is paid off, our balance is zero, and that tells us that tau is equal to 2t. So, we can fill in t and tau in our expression for the equation of value, and finally our accumulation function, because it's 10% annual interest, and unless otherwise specified, we'll assume that's compound interest is going to be 1.1 to power t. Now, to simplify this expression, remember we always assume that a of zero is equal to one, and so this first fraction, a of 2t divided by a of zero, becomes, moreover, the second fraction, a of 2t divided by a of t, becomes, and finally this last fraction is a number divided by itself, so it's just going to be equal to one, and so we get our new equation. Now, we note that 1.1 raised to power 2t, well that's really the same as 1.1 to the t squared, so we can make the substitution x equals 1.1 to the t, x squared is 1.1 to the 2t, and reduce our equation to quadratic, and solve. Now, remember that we're actually solving for 1.1 to power t, and since powers are always positive, we'll only consider the positive root of our quadratic equation, and that turns out to be the time when we add the square root, so hitting both sides with the log and solving gives us t approximately 1.29 years. Suppose you deposit $5,000 at the end of each year for the next three years into an investment account earning 6% annual interest compounded. What transaction must occur after five years if we have 30,000 in the account after 10 years? Here we want the balance to be 30,000 at t equals 10. Our accumulation function is 1.06 to the t, and if we regard the balance as positive, the deposit should also be positive and occur at times t equal 1, 2, and 3. Then there's some transaction p at t equals 5. And so our equation of value will be now while we don't know tau, note that every term has a of tau as a factor so we can divide by it. And we can solve for p. And since we know our accumulation function, we note and we find p is approximately 9,809 and 20 cents. Since deposits are positive, this means we must deposit an additional 9,809 and 20 at t equals 5 to have a final balance of $30,000.