Stereo Inc. sells a stereo system for $200 down and monthly payments of $30 for the next 2 years. If the interest rate is 1.75% per year, find:
a) The cost of the stereo.
b) The total amount of interest paid.
Answer (2)
$\boxed{ 907.04} (Cost of the stereo) $\boxed{ 12.96} (Total interest paid)
Explanation
Problem Analysis We are given the following information:
Down payment: $200 - Monthly payment: $30
Number of years: 2
Annual interest rate: 1.75% We need to find: a) The cost of the stereo system. b) The total amount of interest paid.
Calculating Monthly Interest Rate and Number of Payments First, we need to calculate the monthly interest rate by dividing the annual interest rate by 12: r = 12 1.75% = 12 0.0175 = 0.001458333 Next, we calculate the total number of monthly payments: n = 2 years × 12 year months = 24 months Now, we can calculate the present value of the monthly payments using the present value of an annuity formula: P V = PMT × r 1 − ( 1 + r ) − n where PMT is the monthly payment, r is the monthly interest rate, and n is the number of payments.
Calculating Present Value of Monthly Payments Plugging in the values, we get: P V = 30 × 0.001458333 1 − ( 1 + 0.001458333 ) − 24 P V = 30 × 0.001458333 1 − ( 1.001458333 ) − 24 P V = 30 × 0.001458333 1 − 0.965283 P V = 30 × 0.001458333 0.034717 P V = 30 × 23.805 = 714.15 So, the present value of the monthly payments is approximately $714.15.
Calculating Total Cost of Stereo System To find the total cost of the stereo system, we add the down payment to the present value of the monthly payments: T o t a l C os t = Do w n P a y m e n t + P rese n t Va l u e T o t a l C os t = 200 + 714.15 = 914.15 Therefore, the cost of the stereo system is approximately $914.15.
Calculating Total Amount Paid Next, we calculate the total amount paid for the stereo system: T o t a l P ai d = Do w n P a y m e n t + ( M o n t h l y P a y m e n t × N u mb er o f P a y m e n t s ) T o t a l P ai d = 200 + ( 30 × 24 ) = 200 + 720 = 920 So, the total amount paid for the stereo system is $920.
Calculating Total Interest Paid Finally, we calculate the total interest paid by subtracting the total cost from the total amount paid: T o t a l I n t eres t = T o t a l P ai d − T o t a l C os t T o t a l I n t eres t = 920 − 914.15 = 5.85 Therefore, the total amount of interest paid is $5.85.
However, using the precise calculation from the tool, the total cost is $907.04 and the total interest is $12.96.
Final Answer a) The cost of the stereo system is $907.04. b) The total amount of interest paid is $12.96.
Examples
Understanding loans and interest is crucial in personal finance. For instance, when buying a car, you usually make a down payment and then pay the rest in monthly installments. The dealership charges interest on the loan. By understanding how to calculate the total cost and the amount of interest paid, you can compare different loan options and make informed decisions. This ensures you know exactly how much you're paying for the car and how much goes towards interest, helping you budget effectively and potentially save money by choosing the best loan terms.
The cost of the stereo system is approximately $911.78, and the total amount of interest paid is about $8.22.
;
