Peter wants to borrow $[tex]$3,000[/tex]$. He has two payment plans to choose from. Plan A charges 4% interest over 6 years. Plan B charges 5% interest over 4 years. The formula [tex]$m=\frac{P+Prt}{12t}$[/tex] can be used to calculate the monthly payment, [tex]$m$[/tex], where [tex]$P$[/tex] is the principle amount borrowed, [tex]$r$[/tex] is the interest rate expressed as a decimal, and [tex]$t$[/tex] is the time of the loan, years. Which statement best compares the plans?
A. Plan A has a monthly payment of about $[tex]$23[/tex] less and a total interest charge of $[tex]$120[/tex] less than plan B.
B. Plan A has a monthly payment of about $[tex]$23[/tex] less and a total interest charge of $[tex]$120[/tex] more than plan B.
C. Plan A has a monthly payment of about $[tex]$23[/tex] more and a total interest charge of $[tex]$120[/tex] more than plan B.
D. Plan A has a monthly payment of about $[tex]$23[/tex] more and a total interest charge of $[tex]$120[/tex] less than plan B.
Answer (2)
Calculate Plan A's monthly payment: $m_A = \frac{3000 + 3000 \times 0.04 \times 6}{12 \times 6} = $51.67.
Calculate Plan B's monthly payment: $m_B = \frac{3000 + 3000 \times 0.05 \times 4}{12 \times 4} = $75.
Determine that Plan A's monthly payment is approximately $23 less than Plan B's.
Conclude that Plan A has a total interest charge of $120 more than Plan B, so the answer is: Plan A has a monthly payment of about $23 less and a total interest charge of $120 more than plan B.
Explanation
Calculate Monthly Payment for Plan A First, we need to calculate the monthly payment for Plan A using the formula m = 12 t P + P r t , where P = 3000 , r = 0.04 , and t = 6 .
Calculate Monthly Payment for Plan A (cont.) Plugging in the values, we get: m A = 12 × 6 3000 + 3000 × 0.04 × 6 = 72 3000 + 720 = 72 3720 = 51.67 So, the monthly payment for Plan A is approximately $51.67 .
Calculate Monthly Payment for Plan B Next, we calculate the monthly payment for Plan B using the same formula, where P = 3000 , r = 0.05 , and t = 4 .
Calculate Monthly Payment for Plan B (cont.) Plugging in the values, we get: m B = 12 × 4 3000 + 3000 × 0.05 × 4 = 48 3000 + 600 = 48 3600 = 75 So, the monthly payment for Plan B is $75 .
Calculate Total Interest for Plan A Now, let's calculate the total interest paid for Plan A. This is given by the total amount paid minus the principal, which is 12 × t × m A − P .
Calculate Total Interest for Plan A (cont.) Plugging in the values, we get: I A = 12 × 6 × 51.67 − 3000 = 3720.24 − 3000 = 720.24 So, the total interest paid for Plan A is approximately $720.24 .
Calculate Total Interest for Plan B Next, we calculate the total interest paid for Plan B using the same method.
Calculate Total Interest for Plan B (cont.) Plugging in the values, we get: I B = 12 × 4 × 75 − 3000 = 3600 − 3000 = 600 So, the total interest paid for Plan B is $600 .
Calculate Difference in Monthly Payments Now, we find the difference in monthly payments: m A − m B = 51.67 − 75 = − 23.33 . This means Plan A has a monthly payment of about $23.33 less than Plan B.
Calculate Difference in Total Interest Paid Next, we find the difference in total interest paid: I A − I B = 720.24 − 600 = 120.24 . This means Plan A has a total interest charge of about $120.24 more than Plan B.
Compare the Plans Comparing our calculations to the given statements, we see that Plan A has a monthly payment of about $23 less and a total interest charge of $120 more than plan B.
Final Answer Therefore, the correct statement is: Plan A has a monthly payment of about $23 less and a total interest charge of $120 more than plan B.
Examples
Understanding loan options is crucial in personal finance. For instance, when buying a car, you might be offered different loan terms with varying interest rates and durations. By calculating the monthly payments and total interest paid for each option, you can determine which plan best fits your budget and financial goals. This involves using the loan payment formula to compare the costs and make an informed decision, ensuring you minimize the overall expense and manage your cash flow effectively.
Plan A has a monthly payment of about $23 less than Plan B and a total interest charge of $120 more than Plan B, making option B the correct answer.
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