Find the maturity date and proceeds. Round to the nearest cent.

Face value: $3,500
Discount rate: 9(3/4)%
Date made: July 15
Time: 180 days
Maturity date: underbar(    ?    )
Proceeds: underbar(    ?    )
Select one:

A.
Maturity date: January 11
Proceeds: $3,670.63

B.
Maturity date: January 9
Proceeds: $3,329.38

C.
Maturity date: January 9
Proceeds: $3,670.63

D.
Maturity date: January 11
Proceeds: $3,329.38

To determine the maturity date and proceeds, we'll go through the following steps:

1. **Calculate the Maturity Date**: Add 180 days to the date made (July 15).
2. **Calculate the Proceeds**: Use the discount rate to find the amount received after discounting from the face value.

### Step 1: Calculate the Maturity Date

The date made is July 15. Adding 180 days to July 15 involves:

- July: 31 - 15 = 16 days left in July
- August: 31 days
- September: 30 days
- October: 31 days
- November: 30 days
- December: 31 days

Summing these up:
16 (July) + 31 (August) + 30 (September) + 31 (October) + 30 (November) + 31 (December) = 169 days

Now we need 180 - 169 = 11 more days in January.

Therefore, the maturity date is **January 11**.

### Step 2: Calculate the Proceeds

The face value is $3,500, and the discount rate is 9.75%. The time is 180 days, which is half a year.

The formula for proceeds with a simple interest discount is:

\[ \text{Proceeds} = \text{Face Value} \times \left( 1 - \left( \frac{\text{Discount Rate} \times \text{Time}}{360} \right) \right) \]

Plugging in the values:

\[ \text{Proceeds} = 3500 \times \left( 1 - \left( \frac{0.0975 \times 180}{360} \right) \right) \]

\[ \text{Proceeds} = 3500 \times \left( 1 - \left( \frac{0.0975 \times 0.5} \right) \right) \]

\[ \text{Proceeds} = 3500 \times \left( 1 - 0.04875 \right) \]

\[ \text{Proceeds} = 3500 \times 0.95125 \]

\[ \text{Proceeds} = 3339.375 \]

Rounding to the nearest cent:

\[ \text{Proceeds} = \$3,339.38 \]

### Conclusion

Based on these calculations, the correct answer is:

**D. Maturity date: January 11  
Proceeds: $3,329.38**

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Do you want details or have any questions? Here are 8 related questions for further exploration:

1. How would the proceeds change if the discount rate was 10% instead?
2. What would be the maturity date if the time was 90 days instead of 180 days?
3. How is the discount rate converted to a decimal form in calculations?
4. What would the proceeds be for a face value of $5,000 with the same discount rate and time period?
5. How does the choice of 360 days in the year affect the proceeds calculation?
6. What if the discount rate was compounded monthly instead of being simple interest?
7. How would the maturity date change if the date made was January 1 instead of July 15?
8. Can you explain the concept of discounting in financial terms?

**Tip:** Always double-check the formula used for financial calculations to ensure accuracy.

Learn how to calculate the maturity date and proceeds of a financial instrument using simple interest. This problem involves determining the maturity date and proceeds from a face value of $3,500, a discount rate of 9.75%, and a time period of 180 days. Suitable for advanced high school students and anyone interested in financial calculations.

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