Sure, let's solve this step-by-step.
### Step 1: Understanding Simple Interest
Formula for Simple Interest:
[tex]\[ \text{Simple Interest} = P \times r \times t \][/tex]
Where:
- [tex]\( P \)[/tex] is the principal amount: [tex]$200
- \( r \) is the annual interest rate: 2% or 0.02
- \( t \) is the time in years: 2
Let's plug in the values:
\[ \text{Simple Interest} = 200 \times 0.02 \times 2 \]
\[ \text{Simple Interest} = 200 \times 0.04 = 8 \]
So, Alex earns $[/tex]\[tex]$8$[/tex] in simple interest.
### Step 2: Understanding Compound Interest
Formula for Compound Interest:
[tex]\[ A = P \times (1 + r)^t \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount: [tex]$200
- \( r \) is the annual interest rate: 2% or 0.02
- \( t \) is the time the money is invested for in years: 2
We need to calculate the total amount accumulated (A) and then find the compound interest by subtracting the principal from this amount.
Let's plug in the values:
\[ A = 200 \times (1 + 0.02)^2 \]
\[ A = 200 \times (1.02)^2 \]
\[ A = 200 \times 1.0404 \]
\[ A = 208.08 \]
Compound Interest:
\[ \text{Compound Interest} = A - P \]
\[ \text{Compound Interest} = 208.08 - 200 \]
\[ \text{Compound Interest} = 8.08 \]
So, Chris earns approximately $[/tex]\[tex]$8.08$[/tex] in compound interest.
### Step 3: Calculating the Difference
To determine how much more interest Chris earns compared to Alex, we subtract the simple interest earned by Alex from the compound interest earned by Chris:
[tex]\[ \text{Difference in Interest} = \text{Compound Interest} - \text{Simple Interest} \][/tex]
[tex]\[ \text{Difference in Interest} = 8.08 - 8.00 \][/tex]
[tex]\[ \text{Difference in Interest} = 0.08 \][/tex]
So, Chris earns [tex]$\$[/tex]0.08$ more in interest than Alex over the 2-year period.
