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### Question 1: Compound Interest on a Loan

You take out a five-year loan of £10,000 with a 5% annual compound interest rate.

The compound interest formula is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( A \) is the amount of money accumulated after \( t \) years (including interest).
- \( P \) is the principal amount (initial loan).
- \( r \) is the annual interest rate (as a decimal, so 5% = 0.05).
- \( n \) is the number of times that interest is compounded per year (since the question doesn’t specify, we assume it is compounded annually, so \( n = 1 \)).
- \( t \) is the time the money is invested or borrowed for, in years.

#### Part 1: After Two Years
For the first part, where we need to find the interest after 2 years, we calculate:
\[
A = 10000 \times \left(1 + \frac{0.05}{1}\right)^{1 \times 2}
\]
\[
A = 10000 \times (1.05)^2 = 10000 \times 1.1025 = 11025
\]
The total amount after 2 years is £11,025, and the interest accrued is:
\[
\text{Interest} = A - P = 11025 - 10000 = 1025
\]
So, the interest after two years is **£1,025**.

#### Part 2: After Three Years
For the second part, we calculate the interest after 3 years:
\[
A = 10000 \times \left(1 + \frac{0.05}{1}\right)^{1 \times 3}
\]
\[
A = 10000 \times (1.05)^3 = 10000 \times 1.157625 = 11576.25
\]
The total amount after 3 years is £11,576.25, and the interest accrued is:
\[
\text{Interest} = A - P = 11576.25 - 10000 = 1576.25
\]
So, the interest after three years is **£1,576.25**.

### Question 2: Taxable Interest on Savings

John made £1,243.47 interest on his savings this year. He has a personal savings allowance of £1,000, and any amount above that is taxable at a rate of 20%.

The taxable amount is:
\[
\text{Taxable amount} = 1243.47 - 1000 = 243.47
\]
Now, calculate the tax on the taxable amount:
\[
\text{Tax} = 243.47 \times 0.20 = 48.694
\]
After rounding, the tax is £48.69. Subtract this from the total interest to find John’s post-tax interest:
\[
\text{Post-tax interest} = 1243.47 - 48.69 = 1194.78
\]
So, John makes **£1,194.78** after tax.

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Would you like further details or have any questions?

Here are 5 related questions for practice:
1. What would the interest be after 5 years with the same loan conditions?
2. How much would you need to pay after 3 years if the interest rate was 7% instead of 5%?
3. If John made £800 in interest, would he have to pay any tax at all?
4. What would the post-tax amount be if John had made £1,500 in interest?
5. How much would John make after tax if the taxable rate was increased to 25%?

**Tip:** Always convert percentage rates to decimals when using them in formulas, e.g., 5% becomes 0.05.

I take out a five-year loan of £10,000 with a 5% annual compound interest rate. How much interest would my loan accrue after two years? How much interest would my loan accrue after three years? John made £1,243.47 interest on his savings this year. He has a personal savings allowance of £1,000 but any amount above this is taxable at a rate of 20%. How much interest does John make on his savings after tax?

Learn how to calculate compound interest for a loan of £10,000 with a 5% annual rate over two and three years. This example also covers calculating the tax on savings with a personal allowance of £1,000 and a tax rate of 20%. Suitable for high school students in Grades 9-12.

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