Suppose that you want to deposit $8,000 in an RRSP account that pays interest at a fixed rate of 100r% per year compounded monthly; for example, if the interest rate is 4%, then r = 0.04. Find the rate of interest r on the account if the interest earned after 3 years is $750.
8000((1 + r/12)^(12*4) - 1) = 750
(1 + r/12)^48 = 1.09375
1 + r/12 = 1.00186686
r/12 = 0.00186866
r = 0.0224 = 2 1/4 %
To find the rate of interest (r) on the RRSP account, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final amount after time period t
P = Principal amount (initial deposit)
r = Rate of interest per period
n = Number of compounding periods per year
t = Time period in years
In this case, we are given:
P = $8,000
A = $8,000 + $750 = $8,750 (since the interest earned after 3 years is $750)
t = 3 years
We need to calculate the rate of interest (r).
Now, since the interest is compounded monthly, n = 12 (since there are 12 months in a year).
The formula can be rewritten as:
A = P(1 + r/12)^(12t)
Substituting the given values:
$8,750 = $8,000(1 + r/12)^(12 * 3)
Now we need to solve for r:
(1 + r/12)^(36) = $8,750 / $8,000
(1 + r/12)^(36) = 1.09375
Next, we need to isolate the exponent (36) by taking the logarithm of both sides. We can use either the natural logarithm (ln) or the common logarithm (log).
Taking the natural logarithm (ln) of both sides:
ln((1 + r/12)^(36)) = ln(1.09375)
Using the property ln(a^b) = b * ln(a):
36 * ln(1 + r/12) = ln(1.09375)
Now solve for ln(1 + r/12):
ln(1 + r/12) = ln(1.09375) / 36
Using the inverse natural logarithm (e^x):
1 + r/12 = e^(ln(1.09375) / 36)
Now solve for r:
r/12 = e^(ln(1.09375) / 36) - 1
r = 12 * (e^(ln(1.09375) / 36) - 1)
Using a calculator, we can evaluate the right side of the equation to find the value of r.