 When considering payments at different time periods, we checked our answer by assuming a single payment at a specific time. The idea of paying the entire amount at the beginning leads to the concept of equated time. Could we make all the transactions at a single time and get the same balance? So let's consider. Suppose we deposit an amount equal to the total of all of the deposit amounts. We want to find a time t where the present value of this total is the same as the present value of all the deposits. In other words, if we're assuming compound interest, and we generally are, then a of t is equal to 1 plus i to power t. And using the substitution v equals 1 divided by 1 plus i gives us the reciprocal of a of t is v of t and our equation becomes Now to solve for our time t, we can divide both sides by c, the sum of the contributions, then hit both sides with the log and solve. And we should evaluate this to find the actual equated time. And again, an important idea. Don't memorize formulas. Understand concepts. This formula emerged when we calculated the present value of all of our deposits. Now, while we should evaluate this expression or solve the appropriate equation to find the equated time, it's sometimes useful to use the approximation where the sum ends can be interpreted as the portion of the investment made at time tk weighted by the time tk. And it's doubly important to remember an approximation is not a solution. This approximation tells us about what the solution is, but we should still find it. So for example, suppose we deposit $500 per year for 10 years at 8% annual interest. Let's find and interpret the equated time. So again, there's a formula, but don't memorize formulas. Understand concepts. If we deposit $500 each year for 10 years and earn 8% annual interest, the present value of the investment will be the present value of the $500 invested in year one, plus the present value of $500 invested in year two, and so on. Now, we want this to have the same present value of an investment of $5,000. That's the same as the $10, $500 payments in t years. And so we want the present value to be given by this expression. And since we want our two present values to be equal, we'll equate the two present values to give us an equation. Now, to solve this equation, note that if we let V equals 1 divided by 1.08, then these 1 divided by 1.08 to the nth power can be rewritten. As powers of V. So to simplify, we'll let V equal 1 divided by 1.08, which gives us, we'll divide everything by 5,000, then hit both sides with the log and solve for t. And we evaluate about 5.18 years. And this means our periodic investments are the same as depositing 5,000 now for 5.18 years. Now, we do have that approximate solution. So let's find that approximate solution and verify that it's a good approximation. So as before, we have 10 deposits of 500 for a total deposit of 5,000, the individual deposits are all the same at 500, and the times are from 1 through 10. And so our approximation is, and we find that our time is approximately 5.5. Now, while we could compare the actual value, and we did, we found it was approximately 5.18, this actually defeats the purpose of finding an approximation. Instead, we'll note that 5,000 at 8% interest for 5.5 years grows to, while the actual amount will be, which is close. So remember, there's no real difference mathematically between investing and borrowing, the only real difference is the direction the money travels. So suppose you originally planned to repay a loan by paying 5,005, 10,010, and 25,015. And you decide, hey, I just won the lottery, I want to renegotiate to pay off everything at some point T in the future. So you agree the present value should be based on the annual interest rate of 8%, when should we make a payment? So again, the key idea here is the present value of the payments should be the same. So the present value of the existing payment scheme is, meanwhile, the present value of 40,000 pay that some time T will be, and we want the two present values to be equal. Again, since these powers of 1.08 in the denominator appear quite often, it will be convenient to let that reciprocal be V. So the powers are powers of V, and our equation becomes, we hit both sides with a log and solve. So a single payment of 40,000 in 11.97 years has the same value as the three payments in 5, 10, and 15 years. And finally, we note that an approximate solution will be, or about 12 and a half years. So we did a lot of computations here, but the fact that our approximation is close to our actual solution, we should feel confident in our solution.