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QuestionBetty owes \$57,600 on a 9%, 170-day note. After payments of \$11,520 (day 60) and \$23,040 (day 70), find:
1. Balance after first payment:
2. Balance after second payment:
3. Balance at maturity:

Studdy Solution

STEP 1

Assumptions1. The initial amount owed is 57,600.Theinterestrateis957,600. The interest rate is9%<br />3. The term of the note is170 days4. The first payment of 11,520 is made on day605. The second payment of $23,040 is made on day706. The interest is calculated using the U.S. Rule7. A year is considered to be360 days for the purpose of interest calculation

STEP 2

First, we need to calculate the interest for the first60 days before the first payment is made. The formula for calculating interest isInterest=Principal amounttimesInterest ratetimesTimeInterest = Principal\, amount \\times Interest\, rate \\times Time

STEP 3

Plug in the given values for the principal amount, interest rate, and time to calculate the interest.
Interest=$57,600times9%times60360Interest = \$57,600 \\times9\% \\times \frac{60}{360}

STEP 4

Convert the percentage to a decimal value.
9%=0.099\% =0.09Interest=$57,600times0.09times60360Interest = \$57,600 \\times0.09 \\times \frac{60}{360}

STEP 5

Calculate the interest amount.
Interest=$57,600times0.09times60360=$864Interest = \$57,600 \\times0.09 \\times \frac{60}{360} = \$864

STEP 6

Now that we have the interest amount, we can find the adjusted balance after the first payment. This includes the initial amount owed, the interest, and subtracting the first payment.
Adjusted balance=Principal amount+Interest−First paymentAdjusted\, balance = Principal\, amount + Interest - First\, payment

STEP 7

Plug in the values for the principal amount, the interest, and the first payment to calculate the adjusted balance.
Adjusted balance=$57,600+$864−$11,520Adjusted\, balance = \$57,600 + \$864 - \$11,520

STEP 8

Calculate the adjusted balance after the first payment.
Adjusted balance=$57,600+$864−$11,520=$46,944Adjusted\, balance = \$57,600 + \$864 - \$11,520 = \$46,944

STEP 9

Next, we need to calculate the interest for the next days before the second payment is made. Again, we use the formula for calculating interest, but this time the principal amount is the adjusted balance after the first payment.
Interest=Adjusted balancetimesInterest ratetimesTimeInterest = Adjusted\, balance \\times Interest\, rate \\times Time

STEP 10

Plug in the given values for the adjusted balance, interest rate, and time to calculate the interest.
Interest=$46,944times9%times10360Interest = \$46,944 \\times9\% \\times \frac{10}{360}

STEP 11

Calculate the interest amount.
Interest=$46,944times0.09times10360=$117.36Interest = \$46,944 \\times0.09 \\times \frac{10}{360} = \$117.36

STEP 12

Now that we have the interest amount, we can find the adjusted balance after the second payment. This includes the adjusted balance after the first payment, the interest, and subtracting the second payment.
Adjusted balance=Adjusted balance after first payment+Interest−Second paymentAdjusted\, balance = Adjusted\, balance\, after\, first\, payment + Interest - Second\, payment

STEP 13

Plug in the values for the adjusted balance after the first payment, the interest, and the second payment to calculate the adjusted balance.
Adjusted balance=$46,944+$117.36−$23,040Adjusted\, balance = \$46,944 + \$117.36 - \$23,040

STEP 14

Calculate the adjusted balance after the second payment.
Adjusted balance=$46,944+$117.36−$23,040=$24,021.36Adjusted\, balance = \$46,944 + \$117.36 - \$23,040 = \$24,021.36

STEP 15

Finally, we need to calculate the interest for the remaining100 days to find the balance at maturity. Again, we use the formula for calculating interest, but this time the principal amount is the adjusted balance after the second payment.
Interest=Adjusted balancetimesInterest ratetimesTimeInterest = Adjusted\, balance \\times Interest\, rate \\times Time

STEP 16

Plug in the given values for the adjusted balance, interest rate, and time to calculate the interest.
Interest=$24,021.36times9%times100360Interest = \$24,021.36 \\times9\% \\times \frac{100}{360}

STEP 17

Calculate the interest amount.
Interest=$24,021.36times0.09times100360=$600.54Interest = \$24,021.36 \\times0.09 \\times \frac{100}{360} = \$600.54

STEP 18

Now that we have the interest amount, we can find the balance at maturity. This includes the adjusted balance after the second payment and the interest.
Balance at maturity=Adjusted balance after second payment+InterestBalance\, at\, maturity = Adjusted\, balance\, after\, second\, payment + Interest

STEP 19

Plug in the values for the adjusted balance after the second payment and the interest to calculate the balance at maturity.
Balance at maturity=$24,021.36+$600.54Balance\, at\, maturity = \$24,021.36 + \$600.54

STEP 20

Calculate the balance at maturity.
Balance at maturity=$24,021.36+$600.54=$24,621.90Balance\, at\, maturity = \$24,021.36 + \$600.54 = \$24,621.90The adjusted balance after the first payment is 46,944,afterthesecondpaymentis46,944, after the second payment is 24,021.36, and the balance at maturity is $24,621.90.

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