$1,000,000 \times \frac{\left(\frac{0.07}{12}\right)}{\left(1+\frac{0.07}{12}\right)^{(12 \cdot 29)}-1}
Answer (2)
Calculate the interest rate per month: 12 0.07 ≈ 0.005833 .
Calculate the total number of payments: 12 ⋅ 29 = 348 .
Calculate the monthly interest factor raised to the power of the total number of payments and subtract 1: ( 1 + 12 0.07 ) 348 − 1 ≈ 6.5693 .
Divide the interest rate per month by the result and multiply by the principal amount: 1 , 000 , 000 × 6.5693 0.005833 ≈ 887.97 . The final answer is 887.97 .
Explanation
Understanding the Expression We are asked to evaluate the expression: 1 , 000 , 000 × ( 1 + 12 0.07 ) ( 12 ⋅ 29 ) − 1 ( 12 0.07 ) This expression is used in financial mathematics to calculate loan payments.
Calculating the Interest Rate per Month Let's break down the calculation step by step. First, we calculate 12 0.07 : 12 0.07 ≈ 0.005833
Calculating the Total Number of Payments Next, we calculate the exponent 12 ⋅ 29 : 12 ⋅ 29 = 348
Calculating the Monthly Interest Factor Now, we calculate 1 + 12 0.07 : 1 + 12 0.07 ≈ 1.005833
Calculating the Total Growth Factor Then, we raise this to the power of 348 : ( 1 + 12 0.07 ) 348 ≈ ( 1.005833 ) 348 ≈ 7.5693
Subtracting 1 Next, we subtract 1 from this result: 7.5693 − 1 = 6.5693
Dividing the Interest Rate by the Subtracted Value Now, we divide 12 0.07 by the result: 6.5693 12 0.07 ≈ 6.5693 0.005833 ≈ 0.00088797
Multiplying by the Principal Amount Finally, we multiply this by 1 , 000 , 000 : 1 , 000 , 000 × 0.00088797 ≈ 887.97
Final Answer Therefore, the value of the expression is approximately $887.97.
Examples
This calculation is commonly used to determine the monthly payment required to pay off a loan of $1,000,000 over 29 years with an annual interest rate of 7%. Understanding this formula helps individuals and businesses plan their finances, manage debt, and make informed decisions about borrowing and lending. For example, a homeowner can use this calculation to estimate their monthly mortgage payments, or a business can use it to determine the affordability of a loan for expansion.
The expression evaluates to approximately 887.97, which represents the monthly payment needed to repay a $1,000,000 loan over 29 years at a 7% annual interest rate. We calculated the monthly interest, total number of payments, and applied the formula step-by-step. This helps in understanding loan payments in financial mathematics.
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