Question

Joseph takes out a loan that gathers compound interest.

The table below shows the value of the loan over time.
a) What is the rate of interest per annum?
Give your answer as a percentage to 1 d.p.
b) Work out the value of the loan 10 years after it starts.
Give your answer in pounds (£) to the nearest 1p.

Start
£7500.00

After 1 year
£7965.00

After 2 years
£8458.83

Answer 1

The value of the loan after 10 years is approximately £13686.90

Interest Rate Calculation:

Calculating the interest earned in the first year:

Interest = 7965.00 - 7500.00 = 465.00

Expressing the interest as a percentage of the initial loan amount:

Interest rate = (465.00 / 7500.00) * 100%

Rounding the interest rate to one decimal place:

Interest rate ≈ 6.2%

Therefore, the rate of interest per annum is approximately 6.2%

The loan Value after 10 Years:

Calculating the interest rate per year as a decimal:

Interest rate = 6.2% / 100 = 0.062

using the compound interest formula:

Future value = Present value * (1 + interest rate)^number of years

Future value = 7500.00 * (1 + 0.062)^10

Round the future value to the nearest 1p:

Future value ≈ £13686.90

Therefore, the value of the loan after 10 years is approximately £13686.90

Answer 2

a) 6.2%
b) £13,687

Explanation:

a)

To find the rate of interest, we first find the interest itself.
£7965.00 - £7500.00 = £465.00
Using this, we find the rate of interest in percentage as a fraction of the original loan.

(£7500.00 ÷ £465.0) × 100 = 6.2%

Thus, the rate of interest is 6.2%.

b)
Compound interest is found using the formula

`A = P(1 + (r)/(100))^(t)`
Where

A = final amount

P = initial principal balance

r = interest rate

t = number of time periods elapsed

Thus, with an initial balance of £7500, an interest rate of 6.2% and a time period of 10 years, calculated per annum, we can plug the numbers into the formula, giving us

`A = 7500(1 + (6.2)/(100))^(10)`

`=` £13,686.94
≈ £13,687 (to nearest £1)

Some things to note: number of time periods elapsed mean the amount of times it is compounded; if the question says compounded quarterly for 3 years, t = 12 as it is compounded 4 times a year for 3 years, giving is 12 time periods. This also applies to monthly, bi-weekly etc.