What annual interest rate is required for a debt of $11,385 to grow into $14,383 in 8 years if interest compounds monthly? Round your answer to the nearest tenth of a percent. Question 9 What annual interest rate is required for a debt to grow by 44% in 10 years if interest compounds continuously? Round your answer to the nearest tenth of a percent. Question 10 Suppose that you and your friend both need to borrow the same amount of money. - You borrow money from Bank A. which offers loans at an annual interest rate of 4.8% with continuous compounding. - Your friend borrows money from Bank B, which offers loans an annual interest rate of 3.6% with monthly compounding. If both loans have the same future value and the term of your loan is 94 months, what is the term of your friend's loan (in months)? Round your answer to the nearest month.
For question 8:
We can use the formula A = P(1 + r/n)^(nt), where A is the final amount, P is the initial principal, r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years.
Plugging in the given values, we get:
14,383 = 11,385(1 + r/12)^(12*8)
Solving for r, we get:
r = 0.0574 or 5.74% (rounded to the nearest tenth of a percent)
For question 9:
We can use the formula A = Pe^(rt), where A is the final amount, P is the initial principal, r is the annual interest rate (as a decimal), and t is the number of years.
Since the debt needs to grow by 44%, we know that the final amount will be 1.44 times the initial principal.
Plugging in the given values, we get:
1.44P = Pe^(r*10)
Simplifying, we get:
e^(r*10) = 1.44
Taking the natural logarithm of both sides, we get:
r*10 = ln(1.44)
Solving for r, we get:
r = 0.0421 or 4.2% (rounded to the nearest tenth of a percent)
For question 10:
We can use the formula A = Pe^(rt) to set up two equations, one for each loan, and set them equal to each other since the final amount is the same.
For your loan from Bank A:
A = Pe^(0.048t)
For your friend's loan from Bank B:
A = P(1 + 0.036/12)^(12t/12)
Simplifying, we get:
A = P(1 + 0.003)^t
A = Pe^(0.003t)
Setting these two equations equal to each other, we get:
Pe^(0.048t) = Pe^(0.003t)
Dividing both sides by P and taking the natural logarithm of both sides, we get:
0.048t = 0.003t ln(1.003)
Solving for t, we get:
t = 313 months (rounded to the nearest month)
Question 9:
To find the annual interest rate required for a debt to grow by 44% in 10 years with continuous compounding, we can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A = Final amount
P = Initial amount
r = Annual interest rate (in decimal form)
t = Time in years
e = Euler's number (approximately 2.71828)
Given:
A = 44% = 0.44
P = Initial amount (not given)
t = 10 years
We need to solve for r.
Rearranging the formula:
r = ln(A/P) / t
Plugging in the values:
r = ln(1.44) / 10
r ≈ 0.0492
To convert this to a percentage, multiply by 100:
r ≈ 4.92%
Therefore, an annual interest rate of approximately 4.92% is required for the debt to grow by 44% in 10 years with continuous compounding.
Question 10:
To compare the terms of both loans using different compounding methods, we can use the formula for compound interest:
A = P * (1 + r/n)^(nt)
Where:
A = Final amount
P = Initial amount
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Time in years
For your loan with Bank A:
r = 4.8% = 0.048
n = Compounded continuously
t = 94 months = 94/12 = 7.83 years
For your friend's loan with Bank B:
r = 3.6% = 0.036
n = 12 (monthly compounding)
t = Unknown
Since the final amounts are the same, we can set up an equation to solve for t:
P * e^(0.048*7.83) = P * (1 + 0.036/12)^(12*t)
Canceling out the common terms:
e^(0.048*7.83) = (1 + 0.036/12)^(12*t)
Taking the natural logarithm (ln) of both sides to isolate t:
ln(e^(0.048*7.83)) = ln((1 + 0.036/12)^(12*t))
0.048*7.83 = ln((1 + 0.036/12)^(12*t))
Solving for t:
t ≈ 22.56
Therefore, the term of your friend's loan is approximately 23 months.