Begining loan is 1400 ending amount is 1700 compounded quarterly over 9yrs what is the interest rate?
1700 = 1400 (1+r/4)^(4*9)
r = 2.16%
To find the interest rate on a loan that is compounded quarterly over a specific period of time, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = ending amount (in this case, $1700)
P = beginning loan amount (in this case, $1400)
r = interest rate (to be determined)
n = number of times the interest is compounded per year (in this case, 4 for quarterly)
t = number of years (in this case, 9)
To solve for r, we need to rearrange the formula. First, let's divide both sides of the formula by P:
A/P = (1 + r/n)^(nt)
Next, let's take the natural logarithm (ln) of both sides to eliminate the exponential term:
ln(A/P) = ln[(1 + r/n)^(nt)]
Now we can bring down the exponent using the logarithm property:
ln(A/P) = nt * ln(1 + r/n)
To isolate r, divide both sides of the equation by nt and multiply by n:
ln(A/P) / nt = ln(1 + r/n)
Now, isolate r by multiplying both sides by n and subtracting 1:
n * ln(A/P) / nt - 1 = r/n
Finally, multiply both sides by n to isolate r:
r = n * (ln(A/P) / nt - 1)
Let's substitute the given values into the equation now:
r = 4 * (ln(1700/1400) / (9 * 4) - 1)
Calculating this expression will give us the interest rate (r).