To solve this, let’s go through the calculation step-by-step.

Step 1: Calculate the exact interest

1. Principal ($P$): $2,300
2. Interest Rate ($r$): 9% (or 0.09 as a decimal)
3. Time ($t$): The loan is from September 12 to January 27, which is 137 days.

Using the exact interest formula, which considers a 365-day year:

$\text{Interest} = P \times r \times \frac{t}{365}$

Substituting the values:

$\text{Interest} = 2300 \times 0.09 \times \frac{137}{365}$

Calculating this:

$\text{Interest} = 2300 \times 0.09 \times 0.3753425$ $\text{Interest} \approx 77.27$

4. Maturity Value: This is the total amount Jody will owe on January 27, calculated by adding the interest to the principal.

$\text{Maturity Value} = P + \text{Interest}$ $\text{Maturity Value} = 2300 + 77.27 = 2377.27$

So, the maturity value that Jody will owe on January 27, assuming exact interest, is $2,377.27.

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Step 2: Calculate interest using ordinary interest (assuming a 360-day year)

Using the ordinary interest formula, which considers a 360-day year:

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

Substituting the values:

$\text{Ordinary Interest} = 2300 \times 0.09 \times \frac{137}{360}$

Calculating this:

$\text{Ordinary Interest} = 2300 \times 0.09 \times 0.3805556$ $\text{Ordinary Interest} \approx 78.41$

5. Savings with Exact Interest: This is the difference between the ordinary interest and exact interest.

$\text{Savings} = \text{Ordinary Interest} - \text{Exact Interest}$ $\text{Savings} = 78.41 - 77.27 = 1.14$

So, Jody will save $1.14 in interest with the exact interest method instead of the ordinary interest method.

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Summary:

1. Maturity Value with Exact Interest: $2,377.27
2. Interest Savings with Exact Interest: $1.14

Would you like to know more details on any part of this calculation?

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Related Questions:

1. What would be the maturity value if the interest rate was different, say 10%?
2. How would the savings change if the loan duration was longer or shorter?
3. What effect would a principal amount of $3,000 have on both exact and ordinary interest methods?
4. How would the result change if Jody paid back earlier, say on January 1?
5. How is exact interest used in other types of financial products, like mortgages?

Tip:

When comparing loan terms, always check the type of interest calculation (exact vs. ordinary) as even small differences in days or rates can lead to different costs.