Questions: Comet Lee borrowed 16,000 on a 6%, 90-day note. After 20 days, Comet paid 2,000 on the note. On day 50, Comet paid 4,000 on the note. What are the total interest and ending balance due by the U.S. Rule? Use ordinary interest. Note: Use 360 days. Do not round the denominator in your calculation and round your final answers to the nearest cent.

Comet Lee borrowed 16,000 on a 6%, 90-day note. After 20 days, Comet paid 2,000 on the note. On day 50, Comet paid 4,000 on the note. What are the total interest and ending balance due by the U.S. Rule? Use ordinary interest. Note: Use 360 days. Do not round the denominator in your calculation and round your final answers to the nearest cent.
Transcript text: Comet Lee borrowed $\$ 16,000$ on a $6 \%$, 90-day note. After 20 days, Comet paid $\$ 2,000$ on the note. On day 50, Comet paid $\$ 4,000$ on the note. What are the total interest and ending balance due by the U.S. Rule? Use ordinary interest. Note: Use $\mathbf{3 6 0}$ days. Do not round the denominator in your calculation and round your final answers to the nearest cent.
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Solution

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Solution Steps

Solution Approach

To solve this problem using the U.S. Rule with ordinary interest, follow these steps:

  1. Calculate the initial interest for the first 20 days on the full principal amount.
  2. Subtract the first payment from the principal and calculate the interest for the next 30 days on the new principal.
  3. Subtract the second payment from the principal and calculate the interest for the remaining 40 days on the new principal.
  4. Sum all the interest amounts to get the total interest.
  5. Calculate the ending balance by adding the total interest to the remaining principal after all payments.
Step 1: Calculate Interest for the First 20 Days

The interest for the first 20 days is calculated using the formula: \[ I_1 = P \times r \times \frac{t_1}{d} \] where \( P = 16000 \), \( r = 0.06 \), \( t_1 = 20 \), and \( d = 360 \). Thus, \[ I_1 = 16000 \times 0.06 \times \frac{20}{360} = 53.33 \]

Step 2: Update Principal After First Payment

After the first payment of \( 2000 \), the new principal is: \[ P_1 = P - \text{payment}_1 = 16000 - 2000 = 14000 \]

Step 3: Calculate Interest for the Next 30 Days

The interest for the next 30 days (from day 20 to day 50) is calculated as: \[ I_2 = P_1 \times r \times \frac{t_2}{d} \] where \( t_2 = 30 \). Thus, \[ I_2 = 14000 \times 0.06 \times \frac{30}{360} = 70.00 \]

Step 4: Update Principal After Second Payment

After the second payment of \( 4000 \), the new principal is: \[ P_2 = P_1 - \text{payment}_2 = 14000 - 4000 = 10000 \]

Step 5: Calculate Interest for the Remaining 40 Days

The interest for the remaining 40 days (from day 50 to day 90) is calculated as: \[ I_3 = P_2 \times r \times \frac{t_3}{d} \] where \( t_3 = 40 \). Thus, \[ I_3 = 10000 \times 0.06 \times \frac{40}{360} = 66.67 \]

Step 6: Calculate Total Interest

The total interest is the sum of all interest amounts: \[ \text{Total Interest} = I_1 + I_2 + I_3 = 53.33 + 70.00 + 66.67 = 190.00 \]

Step 7: Calculate Ending Balance

The ending balance due is calculated as: \[ \text{Ending Balance} = P_2 + I_3 = 10000 + 66.67 = 10066.67 \]

Final Answer

The total interest is \( 190.00 \) and the ending balance due is \( 10066.67 \). Therefore, the answers are: \[ \boxed{\text{Total Interest} = 190.00} \] \[ \boxed{\text{Ending Balance} = 10066.67} \]

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