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Question Frank James owes $50,400\$ 50,400 on a 6% note for 150 days. On day 30, he pays $12,600\$ 12,600. On day 100, he pays $17,640\$ 17,640. Calculate the adjusted balances and balance at maturity.

Studdy Solution

STEP 1

Assumptions1. The initial amount owed is 50,400.Theinterestrateis650,400. The interest rate is6%<br />3. The time for repayment is150 days4. The first payment of 12,600 is made on day305. The second payment of $17,640 is made on day1006. The interest is calculated based on the U.S. Rule7. A year is considered to be360 days

STEP 2

First, we need to calculate the interest for the first30 days. We can do this by multiplying the initial amount owed by the interest rate and then multiplying by the number of days divided by360.
Interest30=Initial amounttimesInterest ratetimesDays360Interest_{30} = Initial\, amount \\times Interest\, rate \\times \frac{Days}{360}

STEP 3

Now, plug in the given values for the initial amount, interest rate, and number of days to calculate the interest.
Interest30=$50,400times6%times30360Interest_{30} = \$50,400 \\times6\% \\times \frac{30}{360}

STEP 4

Convert the percentage to a decimal value.
6%=0.066\% =0.06Interest30=$50,400times0.06times30360Interest_{30} = \$50,400 \\times0.06 \\times \frac{30}{360}

STEP 5

Calculate the interest for the first30 days.
Interest30=$50,400times0.06times30360=$252Interest_{30} = \$50,400 \\times0.06 \\times \frac{30}{360} = \$252

STEP 6

Now that we have the interest for the first30 days, we can find the adjusted balance after the first payment. This includes the initial amount, the interest, and subtracting the first payment.
Adjusted balance30=Initial amount+Interest30−First paymentAdjusted\, balance_{30} = Initial\, amount + Interest_{30} - First\, payment

STEP 7

Plug in the values for the initial amount, the interest, and the first payment to calculate the adjusted balance.
Adjusted balance30=$50,400+$252−$12,600Adjusted\, balance_{30} = \$50,400 + \$252 - \$12,600

STEP 8

Calculate the adjusted balance after the first payment.
Adjusted balance30=$50,400+$252−$12,600=$38,052Adjusted\, balance_{30} = \$50,400 + \$252 - \$12,600 = \$38,052

STEP 9

Next, we need to calculate the interest for the next70 days (from day30 to day100). We can do this by multiplying the adjusted balance after the first payment by the interest rate and then multiplying by the number of days divided by360.
Interest70=Adjusted balance30timesInterest ratetimesDays360Interest_{70} = Adjusted\, balance_{30} \\times Interest\, rate \\times \frac{Days}{360}

STEP 10

Now, plug in the given values for the adjusted balance, interest rate, and number of days to calculate the interest.
Interest70=$38,052times6%times70360Interest_{70} = \$38,052 \\times6\% \\times \frac{70}{360}

STEP 11

Calculate the interest for the next70 days.
Interest70=$38,052times0.06times70360=$397.02Interest_{70} = \$38,052 \\times0.06 \\times \frac{70}{360} = \$397.02

STEP 12

Now that we have the interest for the next70 days, 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 balance100=Adjusted balance30+Interest70−Second paymentAdjusted\, balance_{100} = Adjusted\, balance_{30} + Interest_{70} - 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 after the second payment.
Adjusted balance100=$38,052+$397.02−$17,640Adjusted\, balance_{100} = \$38,052 + \$397.02 - \$17,640

STEP 14

Calculate the adjusted balance after the second payment.
Adjusted balance100=$38,052+$397.02−$17,640=$20,809.02Adjusted\, balance_{100} = \$38,052 + \$397.02 - \$17,640 = \$20,809.02

STEP 15

Finally, we need to calculate the interest for the remaining50 days (from day100 to day150). We can do this by multiplying the adjusted balance after the second payment by the interest rate and then multiplying by the number of days divided by360.
Interest50=Adjusted balance100timesInterest ratetimesDays360Interest_{50} = Adjusted\, balance_{100} \\times Interest\, rate \\times \frac{Days}{360}

STEP 16

Now, plug in the given values for the adjusted balance, interest rate, and number of days to calculate the interest.
Interest50=$20,809.02times6%times50360Interest_{50} = \$20,809.02 \\times6\% \\times \frac{50}{360}

STEP 17

Calculate the interest for the remaining50 days.
Interest50=$20,809.02times0.06times50360=$173.41Interest_{50} = \$20,809.02 \\times0.06 \\times \frac{50}{360} = \$173.41

STEP 18

Now that we have the interest for the remaining50 days, we can find the balance at maturity. This includes the adjusted balance after the second payment and the interest.
Balancematurity=Adjusted balance100+Interest50Balance_{maturity} = Adjusted\, balance_{100} + Interest_{50}

STEP 19

Plug in the values for the adjusted balance after the second payment and the interest to calculate the balance at maturity.
Balancematurity=$,809.02+$173.41Balance_{maturity} = \$,809.02 + \$173.41

STEP 20

Calculate the balance at maturity.
Balancematurity=$20,809.02+$173.41=$20,982.43Balance_{maturity} = \$20,809.02 + \$173.41 = \$20,982.43The adjusted balance after the first payment is 38,052,theadjustedbalanceafterthesecondpaymentis38,052, the adjusted balance after the second payment is 20,809.02, and the balance at maturity is $20,982.43.

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