If the tithe is invested at 7.75% APR compounded monthly, it will take around 8.97 years to double in value.
How is compound interest calculated?
FV = P*(1+R/N)(N*T), where FV is the future value, P is the original principle, R is the annual interest rate, N is the number of times interest is compounded annually, and T is the period in years, can be used to calculate the future worth of a loan or investment.
To resolve this issue, we may apply the compound interest formula:
A = P (1 + r/n)
the principal is $35,000, the annual interest rate is 7.75% (or 0.0775 as a decimal)
Since interest is compounded every month, n equals 12.
So we set A = 2P and solve for t:
2P = P (1 + r/n)^(nt)
Dividing both sides by P and simplifying, we get:
2 = (1 + r/n)^(12t)
When we take the natural logarithm of both sides, we obtain:
ln 2 = 12t ln (1 + r/n)
Dividing both sides by 12 ln (1 + r/n), we get:
t = ln 2 / (12 ln (1 + r/n))
Substituting the given values, we get:
t = ln 2 / (12 ln (1 + 0.0775/12))
Using a calculator, we find that t is approximately 8.97 years.
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It will take about 8.9 years for the tithe to double in value if it is invested at 7.75% APR compounded monthly. An annual interest rate of 3.15% compounded semi-annually would be required for $22,500 to accumulate to $50,000 in 14 years. A principal or present value of $27,820.22 would return $40,000 in a Guaranteed Investment Certificate (GIC) in 9 years at 4% APR compounded quarterly.
What is compound interest?
Compound interest is the interest that is calculated not only on the principal amount (the original amount of money), but also on the accumulated interest of previous periods. In other words, it is interest that is earned on interest.
1.To solve this problem, we can use the Time Value of Money (TVM) Solver function to calculate the number of periods required for the investment to double in value.
First, we need to determine the present value (PV) of the tithe, which is $35,000. We also know that the annual percentage rate (APR) is 7.75% and that it is compounded monthly. We can convert the APR to a monthly interest rate by dividing it by 12, which gives us 0.6458% (0.0775 / 12).
Using the TVM Solver function, we can input the following values:
PV = -$35,000 (negative because it represents a cash outflow)
FV = $70,000 (double the initial investment)
i = 0.6458% (monthly interest rate)
n = ? (unknown number of periods)
We want to solve for the number of periods (n) required for the investment to double in value, so we leave that field blank. We also leave the payment (PMT) field blank because there are no regular payments being made.
After inputting these values, we can solve for the number of periods (n) using the TVM Solver function. The result is approximately 107 months, or 8.9 years. Therefore, it will take about 8.9 years for the tithe to double in value if it is invested at 7.75% APR compounded monthly.
2.To solve this problem, we can use the formula for the future value of a lump sum:
[tex]FV = PV x (1 + r/n)^{(n*t)}[/tex]
where FV is the future value, PV is the present value, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.
We can plug in the given values:
PV = $22,500
FV = $50,000
n = 2 (compounded semi-annually, or twice a year)
t = 14 years
We can solve for the annual interest rate (r) by rearranging the formula:
[tex]r = n * [(FV/PV)^{(1/(n*t))} - 1][/tex]
Plugging in the values, we get:
[tex]r = 2 * [(50000/22500)^{(1/(2*14))} - 1][/tex]
r = 3.15%
Therefore, an annual interest rate of 3.15% compounded semi-annually would be required for $22,500 to accumulate to $50,000 in 14 years.
3.We can use the formula for the future value of a lump sum with compound interest to solve for the present value:
[tex]FV = PV x (1 + r/n)^{(n*t)}[/tex]
where FV is the future value, PV is the present value, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.
In this case, we want to solve for PV. We know:
FV = $40,000
r = 4% (0.04) APR, compounded quarterly
n = 4 (compounded quarterly)
t = 9 years
Plugging in the values, we get:
[tex]$40,000 = PV x (1 + 0.04/4)^{(4*9)}[/tex]
Simplifying the right-hand side, we get:
$40,000 = PV x 1.4366
Dividing both sides by 1.4366, we get:
PV = $27,820.22
Therefore, a principal or present value of $27,820.22 would return $40,000 in a Guaranteed Investment Certificate (GIC) in 9 years at 4% APR compounded quarterly.
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