Simplify the equation.
[tex]\begin{array}{l}
\text { PMT }=$81,000 \times \frac{\left(\frac{0.06}{12}\right)}{\left[\left(1+\frac{0.06}{12}\right)^{(12 \cdot 17)}-1\right]} \\
\text { PMT } =$81,000 \times \frac{(\square)}{[(\square)(\square)}-1\right]
\end{array}[/tex]
(Round to seven decimal places as needed.)
Answer (1)
Calculate 0.06/12 = 0.005.
Calculate 1 + 0.06/12 = 1.005.
Calculate (1.005)^(12*17) = (1.005)^204 ≈ 2.7661556.
Calculate PMT = $81,000 * (0.005) / (2.7661556 - 1) = 81 , 000 ∗ ( 0.005 ) /1.7661556. T h es im pl i f i e d e q u a t i o ni s \boxed{PMT = 81,000 \times \frac{(0.005)}{[1.7661556]}} .
Explanation
Understanding the Problem We are given the formula for PMT (periodic payment) and asked to simplify it by filling in the blanks with the appropriate calculations, rounded to seven decimal places. The formula is: PMT = $81,000 \t\times \t\frac{\left(\frac{0.06}{12}\right)}{\left[\left(1+\frac{0.06}{12}\right)^{(12 \cdot 17)}-1\right]} We need to calculate the values inside the parentheses and brackets.
Calculating the numerator First, we calculate 12 0.06 .
12 0.06 = 0.005 Rounded to seven decimal places, this is 0.005.
Calculating the base of the exponent Next, we calculate 1 + 12 0.06 .
1 + 12 0.06 = 1 + 0.005 = 1.005 Rounded to seven decimal places, this is 1.005.
Calculating the exponent Now, we calculate the exponent 12 ⋅ 17 .
12 ⋅ 17 = 204
Calculating the exponential term Then, we calculate ( 1 + 12 0.06 ) ( 12 ⋅ 17 ) = ( 1.005 ) 204 .
( 1.005 ) 204 ≈ 2.7661556 Rounded to seven decimal places, this is 2.7661556.
Subtracting 1 from the exponential term Finally, we subtract 1 from the result: 2.7661556 − 1 = 1.7661556 .
Final Answer Substituting these values into the equation, we get: PMT = $81,000 \t\times \t\frac{(0.005)}{[(2.7661556) - 1]} = $81,000 \t\times \t\frac{0.005}{1.7661556} Thus, the simplified equation is: PMT = $81,000 \t\times \t\frac{(0.005)}{[(1.005)^{(204)}-1]} PMT = $81,000 \t\times \t\frac{(0.005)}{[2.7661556 - 1]} Therefore, the final simplified form is: PMT = $81,000 \t\times \t\frac{(0.005)}{[1.7661556]}
Simplified Equation The simplified equation is:
PMT = $81 , 000 × [ ( 1 + 12 0.06 ) ( 12 ⋅ 17 ) − 1 ] ( 12 0.06 ) PMT = $81 , 000 × [( 1.005 ) ( 204 ) − 1 ] ( 0.005 ) ] PMT = $81 , 000 × [ 2.7661556 − 1 ] ( 0.005 ) ] PMT = $81 , 000 × [ 1.7661556 ] ( 0.005 ) ]
Examples
Understanding loan payments is crucial in personal finance. For instance, when buying a car or a house, the monthly payment (PMT) is determined by the loan amount, interest rate, and loan duration. The formula used here helps calculate that payment. Knowing how to simplify and compute these values allows you to better understand your financial obligations and plan accordingly, ensuring you can manage your debts effectively and make informed decisions about borrowing money.
