$\left(18,500(.04 / 12)(1+.04 / 12)^{\wedge} 60\right) /\left((1+.04 / 12)^{\wedge} 60-1\right)$
Answer (2)
The expression evaluates to approximately 341.23, which represents the monthly payment required to pay off a loan of $18,500 at a 4% annual interest rate over 5 years. This calculation is relevant in financial contexts, particularly for budgeting purposes. Understanding how to compute such payments is essential for effective financial planning.
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Calculate r = 12 0.04 ≈ 0.00333 .
Calculate ( 1 + r ) 60 ≈ 1.2210 .
Calculate the numerator: 18500 ( 1 + r ) 60 ≈ 75.29 .
Calculate the denominator: ( 1 + r ) 60 − 1 ≈ 0.2210 . The final result is 340.71 .
Explanation
Understanding the Problem We are asked to evaluate the expression ( 1 + 12 0.04 ) 60 − 1 18500 ( 12 0.04 ) ( 1 + 12 0.04 ) 60 This expression is likely related to a financial calculation, such as determining the monthly payment on a loan or the present value of an annuity.
Calculating the Expression Let's simplify the expression by breaking it down into smaller parts. Let r = 12 0.04 and n = 60 . The expression becomes ( 1 + r ) n − 1 18500 ( 1 + r ) n First, we calculate r :
r = 12 0.04 = 0.0033333333... Next, we calculate ( 1 + r ) :
1 + r = 1 + 0.0033333333 = 1.0033333333 Then, we calculate ( 1 + r ) n = ( 1 + r ) 60 :
( 1 + r ) 60 = ( 1.0033333333 ) 60 ≈ 1.22099659 Now, we calculate ( 1 + r ) n − 1 = ( 1 + r ) 60 − 1 :
( 1 + r ) 60 − 1 = 1.22099659 − 1 = 0.22099659 Next, we calculate 18500 ( 1 + r ) n = 18500 × 12 0.04 × ( 1 + 12 0.04 ) 60 :
18500 × 0.0033333333 × 1.22099659 ≈ 75.2947899 Finally, we calculate the entire expression: ( 1 + 12 0.04 ) 60 − 1 18500 ( 12 0.04 ) ( 1 + 12 0.04 ) 60 = 0.22099659 75.2947899 ≈ 340.705658 Rounding to two decimal places, we get 340.71.
Final Answer Therefore, the value of the expression is approximately 340.71.
Examples
This calculation is commonly used in finance to determine the monthly payment required to pay off a loan. For instance, if you borrow $18,500 at an annual interest rate of 4% to be paid off over 5 years (60 months), this formula calculates the fixed monthly payment needed. Understanding this formula helps in budgeting and financial planning, ensuring you can manage your loan payments effectively. It's a practical application of exponential growth and division in everyday financial scenarios.
