Juliana invested £5,200 at 6.75% p.a. simple interest. How many days will it take for her investment to grow to at least £5,400?

 days

Round to the next day

To solve this problem, we'll use the formula for simple interest:

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

Where:
- \( A \) is the total amount after interest.
- \( P \) is the principal (initial investment).
- \( r \) is the annual interest rate.
- \( t \) is the time in years.

We are given the following:
- \( P = 5200 \)
- \( A = 5400 \)
- \( r = 6.75\% = 0.0675 \)

We need to determine how many days \( t \) it will take for Juliana's investment to grow to £5,400. First, let's find \( t \) in years.

### Step 1: Set up the equation

Using the simple interest formula:

\[
5400 = 5200(1 + 0.0675t)
\]

### Step 2: Solve for \( t \)

First, divide both sides of the equation by 5200:

\[
\frac{5400}{5200} = 1 + 0.0675t
\]

Simplify the left-hand side:

\[
1.03846 = 1 + 0.0675t
\]

Now, subtract 1 from both sides:

\[
0.03846 = 0.0675t
\]

Next, solve for \( t \):

\[
t = \frac{0.03846}{0.0675} \approx 0.569
\]

### Step 3: Convert years to days

Since \( t = 0.569 \) years, we multiply by 365 to convert this into days:

\[
t_{\text{days}} = 0.569 \times 365 \approx 207.7
\]

Rounding to the next day:

\[
t_{\text{days}} \approx 208 \text{ days}
\]

### Answer: It will take approximately 208 days for Juliana's investment to grow to at least £5,400.

Would you like further details, or do you have any questions?

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Here are 5 related questions to explore further:
1. How would the time change if the interest rate was 5% p.a. instead?
2. What is the total interest earned after 208 days?
3. How would compound interest affect the outcome if calculated annually?
4. If Juliana wanted to reach £5,600 instead, how many days would it take?
5. What is the effective annual rate (EAR) if interest is compounded semi-annually?

**Tip:** Always remember to convert time to appropriate units (days, months, or years) when dealing with interest rates.

Juliana invested £5,200 at 6.75% p.a. simple interest. How many days will it take for her investment to grow to at least £5,400?

Solve Juliana's investment problem using simple interest at 6.75% per annum. Learn how to calculate the time in days needed for the investment to grow from £5,200 to £5,400, using the formula A = P(1 + rt). Suitable for Grades 9-12.

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