$87,000 \times \frac{\left(\frac{0.065}{12}\right)}{\left(1+\frac{0.065}{12}\right)^{(12.8)}-1}
Answer (2)
Calculate the fraction 12 0.065 ≈ 0.005417 .
Calculate the term ( 1 + 12 0.065 ) 12 ⋅ 8 ≈ 1.679669 .
Evaluate the expression ( 1 + 12 0.065 ) 12 ⋅ 8 − 1 12 0.065 ≈ 0.007969566 .
Multiply by 87,000 to get the final answer: 693.35 .
Explanation
Understanding the Problem We are asked to evaluate the expression 87 , 000 × ( 1 + 12 0.065 ) ( 12 ⋅ 8 ) − 1 ( 12 0.065 ) This expression represents the periodic payment of a loan of $87,000 at an annual interest rate of 6.5% compounded monthly for 8 years.
Calculating the Expression Let's break down the calculation into smaller parts. First, we calculate 12 0.065 :
12 0.065 = 0.005416666... ≈ 0.005417 Next, we calculate 1 + 12 0.065 :
1 + 12 0.065 = 1 + 0.005416666... = 1.005416666... ≈ 1.005417 Then, we calculate ( 1 + 12 0.065 ) ( 12 ⋅ 8 ) . Note that 12 ⋅ 8 = 96 :
( 1 + 12 0.065 ) 96 = ( 1.005416666... ) 96 ≈ 1.679669 Now, we calculate ( 1 + 12 0.065 ) 96 − 1 :
( 1 + 12 0.065 ) 96 − 1 = 1.679669 − 1 = 0.679669 Next, we calculate ( 1 + 12 0.065 ) 96 − 1 12 0.065 :
( 1 + 12 0.065 ) 96 − 1 12 0.065 = 0.679669 0.005416666... ≈ 0.007969566 Finally, we calculate 87 , 000 × ( 1 + 12 0.065 ) 96 − 1 12 0.065 :
87 , 000 × 0.007969566 = 693.352237 Therefore, the value of the expression is approximately $693.35.
Final Answer The expression evaluates to approximately 693.35.
Examples
This calculation is used to determine the monthly payment on a loan. For example, if you borrow $87,000 at an annual interest rate of 6.5% compounded monthly and you want to pay it off in 8 years, this formula tells you how much your monthly payment will be. Understanding this calculation helps in financial planning and understanding loan structures.
The expression evaluates to approximately 693.35, which represents the monthly payment for a loan of $87,000 at an interest rate of 6.5% compounded monthly for 8 years. This calculation is important for understanding loan payments. The monthly payment helps in financial planning and budgeting.
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