Jack had a house which got damaged in a storm. He agreed to pay the cost of repairs ($20,000) together with interest in 4 half-yearly instalments of $7,000 each for the repair work. Find the rate of interest he has to pay.

 

This is a typical problem involving instalment payments under compound interest. To find the rate of interest Jack has to pay, we can use the formula for instalment payments under compound interest:

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

Where:
- $ A $ is the amount of each instalment,
- $ P $ is the principal (the total amount to be repaid, which is $20,000),
- $ i $ is the interest rate per period (the unknown we need to solve),
- $ n $ is the number of instalments (4 in this case),
- $ A = 7,000 $ is the amount Jack pays in each instalment.

Substitute the known values into the equation:

\[
7000 = 20000 \times \frac{(1 + i)^4 \times i}{(1 + i)^4 - 1}
\]

We can solve this equation step by step to find $ i $ (the half-yearly interest rate).It appears that the symbolic solution for the interest rate is complex and quite cumbersome to interpret directly in this form. However, we can estimate the interest rate numerically by solving the equation using approximation methods.

Let's compute an approximate solution for the interest rate, $ i $, using numerical methods.The approximate half-yearly interest rate that Jack has to pay is 14.96%.

Would you like more details, or have any questions about this solution?

Here are some related questions you could explore:
1. What would happen to the interest rate if the number of instalments increased to 6?
2. How would the interest rate change if Jack paid $8,000 in each instalment?
3. What is the total interest Jack pays over the course of the four instalments?
4. How would the calculation change if the instalments were monthly instead of half-yearly?
5. What if the principal amount was $25,000 instead of $20,000?

**Tip**: When solving financial problems involving instalments, always double-check whether you're dealing with simple or compound interest, as they require different approaches.

Jack had a house which got damaged in a storm. He agreed to pay the cost of repairs ($20,000) together with interest in 4 half-yearly instalments of $7,000 each for the repair work. Find the rate of interest he has to pay.

Learn how to calculate the interest rate Jack has to pay on a $20,000 repair cost spread over 4 half-yearly instalments of $7,000 each. This problem involves the use of compound interest formulas and numerical methods. Suitable for Grades 11-12.

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