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The problem you've uploaded involves calculating the most cost-effective loan option based on two interest rates for a short-term loan. Here's the breakdown of the question:

### Question Details:

- **Loan Amount**: $700,000
- **Loan Term**: 3 months
- **Option 1**: 4.15% p.a. simple interest
- **Option 2**: 4.9% p.a. interest compounded monthly

The task is to calculate the interest for both options and determine which one William and Kate should choose.

### Approach:

1. **Option 1 (Simple Interest)**:
   The simple interest formula is:
   \[
   I = P \times r \times t
   \]
   Where:
   - $ I $ is the interest
   - $ P = 700,000 $ (the principal)
   - $ r = 0.0415 $ (annual interest rate)
   - $ t = \frac{3}{12} $ years (loan term in years)

2. **Option 2 (Compound Interest)**:
   The compound interest formula is:
   \[
   A = P \left( 1 + \frac{r}{n} \right)^{nt}
   \]
   Where:
   - $ A $ is the amount after interest
   - $ P = 700,000 $ (the principal)
   - $ r = 0.049 $ (annual interest rate)
   - $ n = 12 $ (number of times interest is compounded per year)
   - $ t = \frac{3}{12} $ years

Let's calculate both options.Here are the results for both options:

1. **Option 1 (Simple Interest)**: The total amount to be paid after 3 months is **$707,262.50**.
2. **Option 2 (Compound Interest)**: The total amount to be paid after 3 months is **$708,610.06**.

### Conclusion:
Since Option 1 results in a lower total amount, William and Kate should choose **Option 1 (4.15% p.a. simple interest)**.

Would you like further details or have any questions?

Here are some related questions you might find helpful:
1. How is simple interest different from compound interest in general?
2. What if the loan period were 6 months instead of 3?
3. What if the interest rate for Option 1 were slightly higher, say 4.5% p.a.?
4. How does the frequency of compounding (monthly, quarterly, etc.) affect the total interest?
5. What happens if they decide to repay the loan earlier than 3 months?

**Tip**: When comparing interest rates, always check how frequently the interest is compounded, as it can significantly affect the total amount.

William and Kate are selling their house, but the settlement date is still 3 months away. They will need to take out a bridging loan of $700,000 for 3 months if they are going ahead with the purchase of their dream home by then. Their bank offers two different short-term loan options: Option 1: 4.15% p.a. simple interest. Option 2: 4.9% p.a. interest compounding monthly. Which option should they choose? Use calculations to justify your answer.

This problem compares two loan options for William and Kate, using simple interest and compound interest formulas. The principal amount is $700,000, with a loan term of 3 months. Learn how to calculate both simple and compound interest and determine the best option.

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