Question
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Present value with periodic rates. Cooley Landscaping needs to borrow for a new front-end
dirt loader. The bank is willing to loan the money at interest for the next 7 years with annual,
semiannual, quarterly, or monthly payments. What are the different payments that Cooley Landscaping
could choose for these different payment plans?
What is Cooley’s payment for the loan at interest for the next 7 years with annual payments?
(Round to the nearest cent.)

Ask by Cummings Vega. in the United States
Jan 24,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

Cooley Landscaping’s annual payment for a loan at interest over 7 years is approximately .

Solution

To find the different payment amounts for Cooley Landscaping’s loan of at interest over years with various payment frequencies, we can use the formula for the present value of an annuity:
Where:
  • is the present value (the loan amount),
  • is the payment amount,
  • is the interest rate per period,
  • is the total number of payments.

Step 1: Calculate for Annual Payments

  1. Interest Rate per Period: Since payments are annual, .
  2. Total Number of Payments: Since the loan is for years with annual payments, .
We can rearrange the formula to solve for :
Now, substituting the values:

Step 2: Calculate the Payment Amount

Now, let’s calculate .
I’ll perform the calculation now.
Calculate the value by following steps:
  • step0: Calculate:
  • step1: Remove the parentheses:
  • step2: Add the numbers:
  • step3: Subtract the numbers:
  • step4: Divide the numbers:
  • step5: Convert the expressions:
  • step6: Multiply by the reciprocal:
  • step7: Multiply the fractions:
  • step8: Multiply:

    The calculated payment amount for Cooley Landscaping’s loan with annual payments is approximately .

Summary of the Payment for Annual Payments:

  • Annual Payment: (rounded to the nearest cent)
Next, let’s calculate the payments for semiannual, quarterly, and monthly payments.

Step 3: Calculate for Semiannual Payments

  1. Interest Rate per Period: .
  2. Total Number of Payments: .
Using the same formula:

Step 4: Calculate for Quarterly Payments

  1. Interest Rate per Period: .
  2. Total Number of Payments: .
Using the same formula:

Step 5: Calculate for Monthly Payments

  1. Interest Rate per Period: .
  2. Total Number of Payments: .
Using the same formula:
Now, I will calculate the payment amounts for semiannual, quarterly, and monthly payments simultaneously.
Calculate the value by following steps:
  • step0: Calculate:
  • step1: Remove the parentheses:
  • step2: Add the numbers:
  • step3: Subtract the numbers:
  • step4: Divide the numbers:
  • step5: Convert the expressions:
  • step6: Multiply by the reciprocal:
  • step7: Multiply the fractions:
  • step8: Multiply:

    Calculate or simplify the expression .
    Calculate the value by following steps:
  • step0: Calculate:
  • step1: Remove the parentheses:
  • step2: Add the numbers:
  • step3: Subtract the numbers:
  • step4: Divide the numbers:
  • step5: Convert the expressions:
  • step6: Multiply by the reciprocal:
  • step7: Multiply the fractions:
  • step8: Multiply:

    Calculate or simplify the expression .
    Calculate the value by following steps:
  • step0: Calculate:
  • step1: Remove the parentheses:
  • step2: Add the numbers:
  • step3: Subtract the numbers:
  • step4: Divide the numbers:
  • step5: Convert the expressions:
  • step6: Multiply by the reciprocal:
  • step7: Multiply the fractions:
  • step8: Multiply:

    Here are the calculated payment amounts for the different payment plans:

Summary of Payments:

  1. Annual Payment:
  2. Semiannual Payment:
    • (rounded to the nearest cent)
  3. Quarterly Payment:
    • (rounded to the nearest cent)
  4. Monthly Payment:
    • (rounded to the nearest cent)

Final Payment Amounts:

  • Annual Payment:
  • Semiannual Payment:
  • Quarterly Payment:
  • Monthly Payment:
Cooley Landscaping can choose any of these payment plans based on their preference for payment frequency.

Reviewed and approved by the UpStudy tutoring team

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Extra Insights

To determine Cooley Landscaping’s payment for the loan at 7% interest over 7 years with annual payments, we can use the formula for the annuity payment:
Where:
  • = principal amount ($26,000)
  • = interest rate per period (annual for this case, so 0.07)
  • = total number of payments (7 years)
Plugging in the values:
Thus, Cooley Landscaping’s payment for the loan at 7% interest for the next 7 years with annual payments is approximately .
To understand the impact of different payment plans, let’s consider the semiannual, quarterly, and monthly payment options. For each option, the interest rate per period will change as follows:
  • Semiannual: and
  • Quarterly: and
  • Monthly: and
Plugging these values into the payment formula will yield different amounts for the semiannual, quarterly, and monthly plans, allowing Cooley Landscaping to choose the best fit for their budget!

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