Final answer:
The annual interest rate on George's loan is determined to be 5.4%, calculated from the given values using the compound interest formula. The value of the loan after 10 years, using the same compound interest rate, is approximately £12,974.64, rounded to the nearest penny.
Step-by-step explanation:
To answer the student's questions about the compound interest on George's loan, we first need to determine the annual interest rate and then calculate the future value of the loan after 10 years.
Part a) Determining the Annual Interest Rate
Given the year-over-year increase in the loan amount, we can use the formula for compound interest, which is:
A = P(1 + r)^n
Where:
A = Final amount
P = Initial principal balance (= £7,500)
r = Annual interest rate
n = Number of times the interest is compounded per year
From the table, we have:
Year 1: £7,500 becomes £7,905
Year 2: £7,905 becomes £8,331.87
The interest rate can be found using the following steps:
- Calculate the interest factor for each year by dividing the final amount by the initial principal:
- Year 1: £7,905 / £7,500 = 1.054
- Year 2: £8,331.87 / £7,905 = 1.054
- Find the nth root of the interest factor where n equals the number of years, which in this case is 1, giving us the interest rate for one year.
- 1.054 = (1 + r)^1
- r = 5.4%
Part b) Calculating the Value of the Loan After 10 Years
We can use the compound interest formula again to determine the value of the loan after 10 years:
A = £7,500(1 + 0.054)^{10}
After calculating, the amount would be ~£12,974.64.
Therefore, the future value of the loan after 10 years, rounded to the nearest 1p, is £12,974.64.