To solve this, let's use the simple interest formula:

$A = P + P \cdot R \cdot T$

Where:

• $A$ = Amount after $T$ years
• $P$ = Principal (initial investment)
• $R$ = Rate of interest per year (as a decimal)
• $T$ = Time (in years)

Step 1: Form equations based on given data

1. After 2 years, $A_1 = 5600$: $5600 = P + P \cdot R \cdot 2$

2. After 5 years, $A_2 = 6500$: $6500 = P + P \cdot R \cdot 5$

Step 2: Simplify the equations

1. For $A_1$: $5600 = P (1 + 2R)$

2. For $A_2$: $6500 = P (1 + 5R)$

Step 3: Solve the two equations

Divide the second equation by the first to eliminate $P$: $\frac{6500}{5600} = \frac{1 + 5R}{1 + 2R}$

Simplify: $\frac{13}{11} = \frac{1 + 5R}{1 + 2R}$

Cross-multiply: $13 (1 + 2R) = 11 (1 + 5R)$

Expand: $13 + 26R = 11 + 55R$

Simplify: $2 = 29R$

Solve for $R$: $R = \frac{2}{29} \approx 0.06897 \, \text{or} \, 6.897\%$

Step 4: Find the principal ($P$)

Substitute $R = \frac{2}{29}$ into the first equation: $5600 = P (1 + 2 \cdot \frac{2}{29})$

Simplify: $5600 = P (1 + \frac{4}{29})$ $5600 = P \cdot \frac{33}{29}$

Solve for $P$: $P = \frac{5600 \cdot 29}{33}$ $P = 4927.27$

Final Answer:

• Principal ($P$): $\approx 4927.27$
• Rate of Interest ($R$): $\approx 6.90\%$

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Would you like detailed steps on this, or have further questions?

5 Related Questions:

1. How does the formula for simple interest differ from compound interest?
2. How would the solution change if the interest rate were compounded annually?
3. What is the total interest earned in this scenario over 5 years?
4. Can we solve this using a graphical method? If yes, how?
5. How would this change if the interest was calculated quarterly?

Tip: Always confirm whether interest