SOLUTION: If P dollars are deposited at an interest rate r and compounded n times, the future value An can be found by the formula An=P(1=r)^n. Find the rate of interest if a principal amou

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: If P dollars are deposited at an interest rate r and compounded n times, the future value An can be found by the formula An=P(1=r)^n. Find the rate of interest if a principal amou      Log On


   

Question 628585: If P dollars are deposited at an interest rate r and compounded n times, the future value An can be found by the formula An=P(1=r)^n. Find the rate of interest if a principal amount of $5500 grows to $68990 in 2 years if interest is compunded annually?

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
A%5Bn%5D+=+P%2A%281%2Br%29%5En
You have been given an equation with 4 variables. You have also been given values for 3 of those variables. And you have been asked to find the corresponding 4th value. A basic concept in Math is that if you know all the variables in an equation but one, then you should be able to find the missing value.

Note: I am assuming that the original amount is 55000, not 5500, because 68990 is more than 10 times as large as 5500 and that is an insane increase for just two years of interest. If 5500 is actually correct, then just do the same steps as I will do below, using 5500 instead of 55000.

Inserting the given values into the equation we get:
68990+=+55000%2A%281%2Br%29%5E2
Now we solve for r. First we should simplify. We could multiply out the right side. Or we could divide both sides by 55000. I prefer the latter because it seems easier to multiply out the right side if the 55000 is not there:
68990%2F55000+=%281%2Br%29%5E2
To multiply out the right side we can use FOIL on (1+r)(1+r) or use the %28a%2Bb%29%5E2+=+a%5E2%2B2ab%2Bb%5E2 pattern. I prefer using the pattern:
68990%2F55000+=%281%29%5E2+%2B2%281%29%28r%29+%2B+%28r%29%5E2
which simplifies to:
1.25436364+=+1+%2B2r+%2B+r%5E2

With the equation simplified we can now see that it is a quadratic equation. To solve for r we should make one side zero (by subtracting 1.25436364 from each side)...
0+=+-0.25436364+%2B+2r%2Br%5E2
and then using the Quadratic Formula (the decimal almost means that any other method of solving will be more difficult than the formula):
r+=+%28-%282%29+%2B-+sqrt%28%282%29%5E2-4%281%29%28-0.25436364%29%29%29%2F2%281%29
Simplifying:
r+=+%28-%282%29+%2B-+sqrt%284-4%281%29%28-0.25436364%29%29%29%2F2%281%29
r+=+%28-%282%29+%2B-+sqrt%284-4%281%29%28-0.25436364%29%29%29%2F2%281%29
r+=+%28-%282%29+%2B-+sqrt%284%2B1.01745455%29%29%2F2%281%29
r+=+%28-%282%29+%2B-+sqrt%285.01745455%29%29%2F2%281%29
r+=+%28-2+%2B-+sqrt%285.01745455%29%29%2F2
which is short for:
r+=+%28-2+%2B+sqrt%285.01745455%29%29%2F2 or r+=+%28-2+-+sqrt%285.01745455%29%29%2F2
The second solution will end up being negative. Negative rates make no sense in this word problem so we will reject that one. So the exact rate is
r+=+%28-2+%2B+sqrt%285.01745455%29%29%2F2
This may be an acceptable answer to the problem but probably not. You probably want a decimal (or percent). For this we get out our calculators:
r+=+%28-2+%2B+2.23996753%29%2F2
r+=+%280.23996753%29%2F2
r+=+0.11998376
This is a decimal approximation of the rate. For the percent, as you've probably learned, we just move the decimal over two places:
r+=+11.998376%
So the interest rate is very close to 12%.