 In the real world, you will probably have access to power tools that allow you to solve any equations. But sometimes you don't. For example, on standardized tests like the actuarial FM exam. In that case, you should... Understand how to use the tools you have available. Remember, if the only tool you have is a hammer, then you must treat every problem like a nail. For example, for retirement, Professor Jeff puts aside 2,000 every year into an account earning 3% annual interest. How long will it be before Professor Jeff has 500,000 in the account? So if we deposit 2,000 at t equals 1, 2, and so on, then the total value at t equals n will be... And so we want to solve for n. And we can use our favorite computer algebra system to solve this. Remember, it's your responsibility to use the technology correctly. We don't want imaginary solutions, so we should see if we can find real ones. And since n must be a whole number, and n equals 72 won't give us enough, the solution is n equal to 73. And so it will be 73 years. Maybe we do need those imaginary solutions. Again, there may be situations where for whatever reasons your technology limited. So what if we had to solve this problem using only an annuity calculator? There are many such annuity calculators, some of them online, some of them built into financial calculators, but typically a simple annuity calculator allows you to input the contributions, interest rate, and term to compute the final value. Again, it's your responsibility to use the technology correctly. We've been calculating an annuity immediate, so we need to check that. We know the contribution amount and interest rate, and so we'll guess and check the term. It helps to have an initial guess. For example, if you know the correct answer is one of several possibilities, then any of the possibilities would work as an initial guess. Now, while the real world is often multiple choice, you rarely know what the choices are. So in this case we have to come up with our own initial guess, and so we might start with an initial guess of 500,000 divided by 2,000, that's 250, since that's how long it would take to accumulate $500,000 without considering interest earned. And our initial guess tells us that the amount we have accumulated is too much. Now to get a better answer, we'll use bisection, which is based on the Goldilocks principle. Somewhere between too much and too little is just right. Zero contributions is obviously too little, and 250 is clearly too much, so we'll guess midway between them, about 125, and we find this is too much. To recap, zero is too little, and 125 is too much, so we'll split the difference and guess 62, and find we have not enough. 62 is too little, and 125 is too much, so we guess midway between 62 and 125. We don't actually need to be exactly halfway between, so we can pick a number that's reasonably close to the middle, say 80, and this is still too much. 80 is too much, 62 is too little, so we'll guess about 70, and it turns out that this is not enough, but we're close. So again, 70 is too little, and 80 is too much, so we'll guess 75, and this is still too much, so we guess 72, which is too little, but we do know we're between 72 and 75, so we try 73, which is just a little too much. And again, n equals 72 won't give enough, so it will still be 73 years.