Mike’s Sporting Goods sells elliptical trainers under two payment plans: cash and installment. Under the installment plan, the customer pays $90/month over 5 years with interest changed on the balance at a rate of 19%/year compounded monthly. Find the cash price for an elliptical trainer if it is equivalent to the price paid by a customer using the installment plan. Round your answers to two decimal places.
a. $3,492.47
b. $3,464.42
c. $3,469.47
d. $3,502.80
e. $3,452.28
Please explain thank you.
Mike’s Sporting Goods sells elliptical trainers under two payment plans: cash and installment. Under the installment plan, the customer pays $90/month over 5 years with interest changed on the balance at a rate of 19%/year compounded monthly. Find the cash price for an elliptical trainer if it is equivalent to the price paid by a customer using the installment plan. Round your answers to two decimal places.
a. $3,492.47
b. $3,464.42
c. $3,469.47
d. $3,502.80
e. $3,452.28
Please explain thank you.
To find the cash price for the elliptical trainer, we need to calculate the present value of the installment plan payments.
The formula to calculate the present value of an annuity is:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value (cash price)
PMT = Payment per period ($90/month)
r = Interest rate per period (19% per year compounded monthly = 0.19/12)
n = Number of periods (5 years = 5 * 12 = 60 months)
Now, let's plug in the values into the formula:
PV = 90 * (1 - (1 + (0.19/12))^(-60)) / (0.19/12)
Using a calculator, the cash price is approximately $3,452.28, which is option e.
To find the cash price for an elliptical trainer that is equivalent to the price paid by a customer using the installment plan, we need to use the concept of the present value of future cash flows.
Let's break down the problem into steps:
Step 1: Calculate the monthly interest rate and the number of months of payment.
The annual interest rate is 19%, compounded monthly. So, the monthly interest rate is (19%/12) = 1.58%. The number of months of payment is 5 years * 12 months/year = 60 months.
Step 2: Calculate the monthly payment.
The customer pays $90/month over 5 years, which is 60 months.
Step 3: Calculate the present value of the future payments.
To calculate the present value, we need to discount the future payments using the monthly interest rate.
To simplify the calculation, we can use the formula for the present value of an annuity:
PV = PMT * ((1 - (1 + r)^(-n)) / r)
Where:
PV = Present Value (cash price)
PMT = Monthly Payment
r = Monthly interest rate
n = Number of months of payment
Substituting the values:
PV = 90 * ((1 - (1 + 0.0158)^(-60)) / 0.0158)
Step 4: Calculate the cash price by rounding the calculated present value to two decimal places.
PV ≈ 3462.94, rounded to two decimal places
Therefore, the cash price for an elliptical trainer equivalent to the price paid by a customer using the installment plan is approximately $3,462.94.
The closest option provided is b. $3,464.42, which is the correct answer.