Karina paid for a living room suite over time, signing an instalment contract that requires semi-annual payments of $2560.00 for 8.5 years. The first payment is made on the signing date and interest is 14.5% compounded semi-annually. What was the cash price? (Hint: Simply calculate the PV of this simple annuity due)

To determine the cash price of the living room suite, we need to calculate the present value (PV) of the simple **annuity due**. Since the payments are made semi-annually and the first payment is made immediately (at the time of signing), it's considered an annuity due.

### Key information:
- **Semi-annual payment** = $2560.00
- **Time period** = 8.5 years (semi-annual periods = $8.5 \times 2 = 17 \text{ periods}$)
- **Interest rate** = 14.5% compounded semi-annually (semi-annual interest rate = $ \frac{14.5}{2} = 7.25\%$)
- **Annuity type**: Annuity due (payments are made at the beginning of each period)

#### Formula for Present Value of an Annuity Due:

\[
PV = P \times \left( \frac{1 - (1 + i)^{-n}}{i} \right) \times (1 + i)
\]

Where:
- $PV$ = Present Value (the cash price we are calculating)
- $P$ = Payment per period ($2560)
- $i$ = Interest rate per period (7.25% or 0.0725)
- $n$ = Number of periods (17 periods)

Let's plug in the values and calculate the present value.The cash price of the living room suite, based on the present value of the annuity due, is approximately **$26,347.89**.

Would you like more details or have any further questions?

Here are five related questions to explore this concept further:
1. How would the cash price change if the interest rate were 10% compounded semi-annually?
2. What would the cash price be if the payments were made annually instead of semi-annually?
3. How would the total amount paid by Karina change if the term was extended to 10 years?
4. What is the total interest paid over the 8.5 years?
5. How does the concept of annuity due differ from an ordinary annuity in terms of payment timing?

**Tip:** Annuities due typically result in a higher present value than ordinary annuities because payments are made at the start of each period, meaning they accrue more interest.

Learn how to calculate the present value of an annuity due for Karina's semi-annual payments of $2560 over 8.5 years at 14.5% interest compounded semi-annually. This solution explains the key concepts of annuities, interest rates, and compound interest, providing a step-by-step breakdown of the formula. Suitable for high school students.

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