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Let's break down the problem and solve it step by step.

**Problem Summary:**

- Geoff and Jeff both invest £x for 5 years.
- Geoff’s bank offers 5% simple interest.
- Jeff’s bank offers 4% compound interest.
- After 5 years, Jeff's investment is worth £243.13 more than Geoff's investment.
- We need to find the value of **x** (the initial investment), to the nearest pound.

---

### Step 1: Calculate Geoff's final investment (Simple Interest)

The formula for **Simple Interest** is:

\[
A = P(1 + rt)
\]

Where:
- \(A\) is the final amount
- \(P\) is the initial principal (which is \(x\) in this case)
- \(r\) is the annual interest rate (5% = 0.05 for Geoff)
- \(t\) is the time in years (5 years here)

So Geoff’s final amount is:

\[
A_G = x(1 + 0.05 \times 5) = x(1 + 0.25) = 1.25x
\]

---

### Step 2: Calculate Jeff's final investment (Compound Interest)

The formula for **Compound Interest** is:

\[
A = P(1 + \frac{r}{n})^{nt}
\]

Where:
- \(P\) is the initial principal (which is \(x\) in this case)
- \(r\) is the annual interest rate (4% = 0.04 for Jeff)
- \(n\) is the number of times the interest is compounded per year (assuming annually, \(n = 1\))
- \(t\) is the time in years (5 years here)

So Jeff’s final amount is:

\[
A_J = x(1 + \frac{0.04}{1})^{1 \times 5} = x(1.04)^5
\]

Using the approximate value of \(1.04^5\):

\[
A_J = x \times 1.2166529
\]

---

### Step 3: Set up the equation based on the difference

We are told that after 5 years, Jeff’s investment is worth £243.13 more than Geoff’s:

\[
A_J - A_G = 243.13
\]

Substitute the expressions for \(A_J\) and \(A_G\):

\[
x(1.2166529) - x(1.25) = 243.13
\]

Simplify the equation:

\[
x(1.2166529 - 1.25) = 243.13
\]

\[
x(-0.0333471) = 243.13
\]

Now, solve for \(x\):

\[
x = \frac{243.13}{-0.0333471}
\]

I'll calculate this value for you.

The calculated value of \(x\) is approximately \(-£7290.89\), which doesn't seem reasonable since a negative investment isn't possible. It seems I made an error in interpreting the difference.

Let's re-evaluate the situation by flipping the equation. Jeff’s investment should be more than Geoff's, so the correct equation should be:

\[
A_J - A_G = 243.13
\]

Where \( A_J \) (Jeff's final investment) is **more** than \( A_G \). I'll correct the math and solve it again.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to walk you through the steps to solve it manually, or help with another question?

Here are 5 related questions for further practice:
1. How would the formula change if Geoff’s investment had compound interest instead?
2. What would the answer be if the time period was extended to 10 years instead of 5?
3. How would you calculate the interest rate if you knew the initial and final values?
4. How does the difference between simple and compound interest grow over time?
5. How would the solution change if the interest was compounded quarterly?

**Tip:** Compound interest grows exponentially, so even small differences in the rate can make a big impact over time!

Geoff and Jeff both invest £x for 5 years. Geoff invests in a bank paying 5% simple interest and Jeff in a bank paying 4% compound interest. After 5 years Jeff’s investment is worth £243.13 more than Geoff’s. Find the value of x, to the nearest £.

Solve the problem where Geoff invests at 5% simple interest and Jeff at 4% compound interest over 5 years. Jeff’s investment exceeds Geoff's by £243.13 after 5 years. Learn how to calculate the initial investment (x) using simple and compound interest formulas. Suitable for Grades 9-11.

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