Certainly! Let's break down the steps to find the amount of money that Charlie must pay back after taking out a loan of [tex]$20,000 with a compound interest rate of $[/tex]4\%$ per year for 3 years.
1. Principal (P): The initial amount of money borrowed. For this loan, \( P = £ 20,000 \).
2. Rate (r): The annual interest rate. In this case, \( r = 4\% \) per year. To use this rate in calculations, we convert it to a decimal: \( r = \frac{4}{100} = 0.04 \).
3. Time (t): The number of years the money is borrowed for. Here, \( t = 3 \) years.
4. Compounding Periods (n): Since the interest is compounded annually, \( n = 1 \).
The compound interest formula to calculate the amount (A) to be paid back is given by:
[tex]\[ A = P \times \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Substituting the values we have:
[tex]\[ A = 20,000 \times \left(1 + \frac{0.04}{1}\right)^{1 \times 3} \][/tex]
[tex]\[ A = 20,000 \times (1 + 0.04)^3 \][/tex]
[tex]\[ A = 20,000 \times 1.04^3 \][/tex]
By solving this expression:
1. The base of our expression (the expression inside the parentheses) is \( 1.04 \).
2. The exponent (to which the base is raised) is \( 3 \).
Thus, the completed expression for the amount of money that Charlie must pay back after 3 years is:
[tex]\[ £ 20,000 \times 1.04^3 \][/tex]
Evaluating \( 1.04^3 \):
[tex]\[ 1.04^3 \approx 1.124864 \][/tex]
And calculating the final amount:
[tex]\[ A \approx 20,000 \times 1.124864 \][/tex]
[tex]\[ A \approx £ 22497.28 \][/tex]
Therefore, the amount of money that Charlie must pay back at the end of 3 years is approximately £22,497.28.
