Question 628585
{{{A[n] = P*(1+r)^n}}}
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.<br>
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.<br>
Inserting the given values into the equation we get:
{{{68990 = 55000*(1+r)^2}}}
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:
{{{68990/55000 =(1+r)^2}}}
To multiply out the right side we can use FOIL on (1+r)(1+r) or use the {{{(a+b)^2 = a^2+2ab+b^2}}} pattern. I prefer using the pattern:
{{{68990/55000 =(1)^2 +2(1)(r) + (r)^2}}}
which simplifies to:
{{{1.25436364 = 1 +2r + r^2}}}<br>
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 + 2r+r^2}}}
and then using the Quadratic Formula (the decimal almost means that any other method of solving will be more difficult than the formula):
{{{r = (-(2) +- sqrt((2)^2-4(1)(-0.25436364)))/2(1)}}}
Simplifying:
{{{r = (-(2) +- sqrt(4-4(1)(-0.25436364)))/2(1)}}}
{{{r = (-(2) +- sqrt(4-4(1)(-0.25436364)))/2(1)}}}
{{{r = (-(2) +- sqrt(4+1.01745455))/2(1)}}}
{{{r = (-(2) +- sqrt(5.01745455))/2(1)}}}
{{{r = (-2 +- sqrt(5.01745455))/2}}}
which is short for:
{{{r = (-2 + sqrt(5.01745455))/2}}} or {{{r = (-2 - sqrt(5.01745455))/2}}}
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 = (-2 + sqrt(5.01745455))/2}}}
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 = (-2 + 2.23996753)/2}}}
{{{r = (0.23996753)/2}}}
{{{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%.