SOLUTION: What annual interest rate would you need to earn if you wanted a $600 per month contribution to grow to $46,000 in six years? (Do not round intermediate calculations. Round your fi

Question 1093011: What annual interest rate would you need to earn if you wanted a $600 per month contribution to grow to $46,000 in six years? (Do not round intermediate calculations. Round your final answer to 2 decimal places.)
Answer by greenestamps(13209) (Show Source): You can put this solution on YOUR website!

The formula for the accumulated value V of an investment of amount A deposited monthly for n months at an annual interest rate of r is

You are looking for the annual interest rate r which will produce value V = 46000 with deposit amount A = 600 in 6 years, or 72 months, so n = 72.

There are 4 parameters here: V, A, n, and r. If you know r and any two of the others, you can find the missing one algebraically. However, if you know V, A, and n and are looking for r, then there is no algebraic way to solve for it. That is essentially because r occurs in two different places in the formula that make it impossible to solve the formula for r.

So you need to use numerical methods. In your problem we need to find r knowing

We can divide by the 600 to give us

Then we can use a graphing calculator to graph the constant on the left and the expression on the right to find where the two graphs intersect. The value I got for r/12 (to several decimal places) is .00175194. Since that is the monthly interest rate, the annual interest rate is 12 times that number, which is .02102328. As a percentage rounded to 2 decimal places, that is 2.10%.

Since that rounded percentage rate is slightly less than the actual percentage rate, plugging 2.10% into the formula shown at the top of my response gives a value that is a few dollars short of the target of $46,000.

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