To find the maturity value of the loan using the ordinary interest method (which assumes a 360-day year), we will follow these steps:

1. Identify the principal (loan amount), interest rate, and time period:

   • Principal (P) = $33,000
   • Interest rate (r) = 6% or 0.06
   • Loan period: From May 17 to January 16 of the following year.

2. Calculate the loan period in days:

   • From May 17 to January 16:
     • May 17 to May 31: 14 days
     • June: 30 days
     • July: 31 days
     • August: 31 days
     • September: 30 days
     • October: 31 days
     • November: 30 days
     • December: 31 days
     • January 1 to January 16: 16 days

   Total days = 14 + 30 + 31 + 31 + 30 + 31 + 30 + 31 + 16 = 244 days

3. Use the formula for ordinary interest:

   $\text{Interest} = P \times r \times \frac{t}{360}$

   Where:

   • $P = 33,000$
   • $r = 0.06$
   • $t = 244$ days

   Substituting these values:

   $\text{Interest} = 33,000 \times 0.06 \times \frac{244}{360}$

4. Calculate the interest:

   $\text{Interest} = 33,000 \times 0.06 \times 0.6778 \approx 1,344.60$

5. Calculate the maturity value:

   $\text{Maturity Value} = \text{Principal} + \text{Interest} = 33,000 + 1,344.60 = 34,344.60$

Final Answer:

The maturity value due on January 16 is $34,344.60.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What would the maturity value be if the loan had an interest rate of 7%?
2. How would the maturity value change if the loan period was extended by one month?
3. What is the difference between ordinary interest and exact interest methods?
4. How would the calculation change if the year was not considered 360 days?
5. How does compounding interest affect the maturity value of a loan?

Tip: Always double-check the time period when calculating interest to ensure accuracy, especially when using ordinary interest which assumes a 360-day year.