An article cost $29885. To buy this article, a down payment of $3420 is needed.If interest charged is 16% compunded quarterly, how much should be paid at the end of every 3 months for 1 year in order to payoff the balance?
please explain,i don't understand...thank you very much
This is what is called an amortization problem. It is usually solved with a table or an iterating program. Here is a way to get the approximate answer directly. The principal due at the beginning is 29885-3420 = $26465. Over one year, with a declining balance, the average balance is $13232 and the interest on that amount over one year will be $2117.
It the loan is paid off in equal quarterly amounts, they must add up to 26465 + 2117 = $28582. The quarterly paymewnts should be 1/4 of that, or $7146.
Check:
Balance due at start (after down payment):
26465
After one quarter: add 1058.60 for interest due and subtract 7146 principal payment. Remaining balance = 20,377.60
After first quarter: add $815.10 interest and subract 7146 principl. Remaining balance = 14,046.70
After third quarter: add 561.87 interest and subtract 7146. Remaining balance = $7462.57
After fourth quarter: add $298.50 interest and subtract 7146. Remaining balance = $615.
For a second iteration, I would recommend adding $154 to each quarterly ayment, to get rid of the $615 deficit on the first attempt.
That makes the quarterly payment $7300 after one iteration. I end up overpaying $38 this way, so the third iterated answer is $10 less per quarter, or $7290.
There is a handy amortization calculator at this web site:
http://www.yona.com/loan/
Enter the initial loan balance of 26465, the 16$ interest rate, and 4 quarterly payments.
Using it, with quarterly compounding you should get an exact loan payment of $7290.84
27523.60
9765.25
To calculate the amount that needs to be paid at the end of every 3 months in order to pay off the balance, we need to consider the initial cost of the article, the down payment, the interest rate, and the compounding period.
First, let's calculate the remaining balance after the down payment is made. We subtract the down payment from the initial cost:
Remaining balance = Initial cost - Down payment
Remaining balance = $29,885 - $3,420
Remaining balance = $26,465
Next, we need to calculate the interest on the remaining balance. The interest is compounded quarterly, which means it is applied every 3 months. The interest rate is 16% per year, compounded quarterly. To calculate the quarterly interest rate, we divide the annual rate by 4 (since there are 4 quarters in a year):
Quarterly interest rate = Annual interest rate / Number of compounding periods in a year
Quarterly interest rate = 16% / 4
Quarterly interest rate = 0.16 / 4
Quarterly interest rate = 0.04 or 4%
Now, we can use the formula for compound interest to calculate the remaining balance after each quarter:
Remaining balance after each quarter = Remaining balance * (1 + Quarterly interest rate)
For simplicity, let's calculate the quarterly balance for 1 year, which is equivalent to 4 quarters. By putting the values into the formula, we can calculate:
Quarter 1 balance = $26,465 * (1 + 0.04)
Quarter 1 balance = $26,465 * 1.04
Quarter 1 balance ≈ $27,549
Quarter 2 balance = Quarter 1 balance * (1 + 0.04)
Quarter 2 balance ≈ $27,549 * 1.04
Quarter 2 balance ≈ $28,631
Quarter 3 balance = Quarter 2 balance * (1 + 0.04)
Quarter 3 balance ≈ $28,631 * 1.04
Quarter 3 balance ≈ $29,731
Quarter 4 balance = Quarter 3 balance * (1 + 0.04)
Quarter 4 balance ≈ $29,731 * 1.04
Quarter 4 balance ≈ $30,850
Therefore, at the end of each quarter, approximately the following amounts should be paid in order to pay off the balance:
Quarter 1 payment = $27,549 - $26,465
Quarter 1 payment ≈ $1,084
Quarter 2 payment = $28,631 - $27,549
Quarter 2 payment ≈ $1,082
Quarter 3 payment = $29,731 - $28,631
Quarter 3 payment ≈ $1,100
Quarter 4 payment = $30,850 - $29,731
Quarter 4 payment ≈ $1,119
So, to pay off the balance over 1 year, you would need to make payments of approximately $1,084 at the end of each quarter.