Suppose Fred borrowed $5,616 for 25 months and Joanna borrowed $4,095.
Fred's loan used the simple discount model with an annual rate of 8.1% while Joanne's loan used the simple interest model with an annual rate of 3.4%.
If their maturity values were the same, how many months was Joanna's loan for?
To find out how many months Joanna's loan was for, we need to compare the maturity values of Fred's and Joanna's loans.
The maturity value of a loan can be calculated using the formulas for the simple discount model and the simple interest model, respectively.
For Fred's loan:
Maturity Value = Principal - Discount
Maturity Value = $5,616 - Discount
For Joanna's loan:
Maturity Value = Principal + Interest
Maturity Value = $4,095 + Interest
Given that the maturity values for both loans are the same, we can set up an equation:
$5,616 - Discount = $4,095 + Interest
Now let's calculate the Discount and Interest for each loan.
For Fred's loan, we need to determine the Discount:
Discount = Principal * Rate * Time
Given that the Principal is $5,616 and the Rate is 8.1% per year, we need to convert the Rate to a monthly rate by dividing it by 12:
Monthly Rate = 8.1% / 12 = 0.675%
Now we can calculate the Discount for Fred's loan:
Discount = $5,616 * 0.675% * 25 months
For Joanna's loan, we need to determine the Interest:
Interest = Principal * Rate * Time
Given that the Principal is $4,095 and the Rate is 3.4% per year, we need to convert the Rate to a monthly rate by dividing it by 12:
Monthly Rate = 3.4% / 12 = 0.2833%
Now we can calculate the Interest for Joanna's loan:
Interest = $4,095 * 0.2833% * Time
Now, we can substitute the calculated values into the equation:
$5,616 - Discount = $4,095 + Interest
Solving this equation will give us the value of Time, which represents the number of months for Joanna's loan.
To find out how many months Joanna's loan was for, we need to set up an equation using the formulas for the maturity values of the loans.
For Fred:
M = P - (P * r * t)
where M is the maturity value, P is the principal amount (borrowed amount), r is the annual interest rate, and t is the time period in years.
For Joanna:
M = P + (P * r * t)
where M is again the maturity value, P is the principal amount, r is the annual interest rate, and t is the time period in years.
We are given that the maturity values are equal, so we can set up the equation:
Pf - (Pf * rf * tf) = Pj + (Pj * rj * tj)
where Pf is the principal amount of Fred's loan, rf is the annual interest rate of Fred's loan, tf is the time period in years for Fred's loan, Pj is the principal amount of Joanna's loan, rj is the annual interest rate of Joanna's loan, and tj is the time period in years for Joanna's loan.
Substituting the given values:
5616 - (5616 * 0.081 * 25) = 4095 + (4095 * 0.034 * tj)
Calculating the left side:
5616 - (4544.16)= 4095 + (139.17 * tj)
Simplifying:
1071.84 = 4095 + (139.17 * tj)
Now, we can solve for tj:
1071.84 - 4095 = 139.17 * tj
-3023.16 = 139.17 * tj
Dividing both sides by 139.17:
tj = -3023.16 / 139.17
tj ≈ -21.73
The negative value for tj doesn't make sense in this context. It indicates that Joanna's loan term is not a valid positive number of months. Please double-check the given information or calculations to ensure accuracy.