2. You charged $ 1,000 on your credit card for Christmas presents. Your credit card company charges you 16 % annual interest, compounded monthly. If you make the minimum payments of $25 per month, how long will it take [to the nearest month] top pay off your balance?
1000 = 25[1 - 1.0133333^-n]/.0133333
.5333333 = 1 - 1.0133333^-n
.4666667 = 1.0133333^-n
-n = log .46666667/log1.0133333
n = 57.54 payments
there will be 57 payments of $25 plus a partial payment less than $25 on the 58th month
To calculate the time it will take to pay off your credit card balance, we can use the formula for the number of periods it takes to reach a specific future value.
First, let's calculate the monthly interest rate. We know that the annual interest rate is 16%, so the monthly interest rate would be 16% divided by 12 months, which equals 1.333%.
Next, let's determine the future value, which is the outstanding balance on your credit card. Initially, the balance is $1,000.
Now, we can calculate the number of periods using the formula:
n = - log(1 - r * PV / PMT) / log(1+r)
Where:
n = number of periods (months)
r = monthly interest rate
PV = present value (outstanding balance)
PMT = monthly payment
Substituting the known values into the formula:
n = - log(1 - 0.01333 * 1000 / 25) / log(1 + 0.01333)
Using a calculator:
n ≈ - log(1 - 0.0266) / log(1.01333)
n ≈ - log(0.9734) / log(1.01333)
n ≈ 32.57
Rounding to the nearest month, it will take approximately 33 months to pay off the balance on your credit card.
To calculate how long it will take to pay off your credit card balance, we need to use the formula for the number of months it takes to pay off a loan when making consistent monthly payments.
First, let's calculate the monthly interest rate. Since the annual interest rate is 16%, we divide it by 12 to get the monthly interest rate: 16% / 12 = 0.16 / 12 = 0.0133 (approximately).
Now, let's determine the monthly payment and the remaining balance after each payment. We'll use a loop to keep subtracting the monthly payment from the balance and adding the monthly interest until the balance reaches zero.
Starting with a balance of $1,000, we'll subtract the monthly payment of $25 and then add the monthly interest to the remaining balance each month.
Here's how the calculation would look like:
Month 1:
Balance: $1,000
Interest: 0.0133 * $1,000 = $13.30
Payment: $25
Remaining Balance: $1,000 + $13.30 - $25 = $988.30
Month 2:
Balance: $988.30
Interest: 0.0133 * $988.30 = $13.14
Payment: $25
Remaining Balance: $988.30 + $13.14 - $25 = $976.44
By repeating this process, we can determine the number of months it will take to pay off the balance. When the remaining balance reaches zero, we have our answer.
Calculating the remaining balance for subsequent months:
Month 3:
Remaining Balance: $964.82
Month 4:
Remaining Balance: $953.38
Month 5:
Remaining Balance: $942.13
Month 6:
Remaining Balance: $931.05
...
Month 35:
Remaining Balance: $3.89
Month 36:
Remaining Balance: $0
From the calculation, we see that it takes 36 months (or 3 years) to completely pay off the credit card balance when making minimum payments of $25 per month.