Let's solve the problems from Exercise 1 and 2 on the sheet:

Exercise 1: Simple Interest Problems

1. Question 1: For how many years should a person invest ₹10,000 to get a simple interest of ₹2,000 @ 5% p.a.?

   Solution: The formula for Simple Interest (SI) is: $SI = \frac{P \times R \times T}{100}$ where:

   • $P = 10,000$ (Principal)
   • $R = 5\%$ (Rate of interest)
   • $SI = 2,000$ (Simple Interest)

   Plugging in the values, we solve for $T$ (Time): $2,000 = \frac{10,000 \times 5 \times T}{100}$ Simplifying: $2,000 = 500 \times T$ $T = \frac{2,000}{500} = 4$ So, the answer is 4 years.

2. Question 2: At what rate of interest will a person earn a simple interest of ₹1,800 on a sum of ₹15,000 for three years?

   Solution: Using the SI formula: $SI = \frac{P \times R \times T}{100}$ Given:

   • $P = 15,000$
   • $SI = 1,800$
   • $T = 3$ years

   Plugging in the values to solve for $R$: $1,800 = \frac{15,000 \times R \times 3}{100}$ Simplifying: $1,800 = 450 \times R$ $R = \frac{1,800}{450} = 4\%$ So, the rate of interest is 4%.

Exercise 2: Compound Interest Problems

1. Question 6: For how many years should a person invest ₹4,000 @ 5% p.a. to get an interest of ₹410, under compound interest, compounded annually?

   Solution: The formula for Compound Interest (CI) is: $A = P \left(1 + \frac{R}{100}\right)^T$ where:

   • $P = 4,000$
   • $R = 5\%$
   • $A - P = 410$ (CI)

   First, find the total amount $A$: $A = 4,000 + 410 = 4,410$

   Plugging in the values to solve for $T$: $4,410 = 4,000 \left(1 + \frac{5}{100}\right)^T$ $4,410 = 4,000 \times \left(1.05\right)^T$ Dividing both sides by 4,000: $1.1025 = (1.05)^T$ Solving for $T$ using logarithms: $T = \log_{1.05}(1.1025)$ Approximately, $T = 2$ years.

2. Question 7: At what rate of interest will a person earn an interest of ₹4,641 on a sum of ₹10,000 for four years, interest being compounded annually?

   Solution: Given:

   • $P = 10,000$
   • CI = 4,641
   • $T = 4$ years Total amount $A$: $A = 10,000 + 4,641 = 14,641$

   Using the CI formula: $14,641 = 10,000 \left(1 + \frac{R}{100}\right)^4$ Simplifying: $1.4641 = \left(1 + \frac{R}{100}\right)^4$ Taking the fourth root: $1 + \frac{R}{100} = 1.1$ $R = 10\%$ So, the rate of interest is 10%.

Would you like further explanations or help with other questions?

Follow-up Questions:

1. How is the formula for compound interest derived?
2. How does the interest rate affect the total amount in compound interest compared to simple interest?
3. What are the differences between simple and compound interest?
4. How can logarithms be used to find the time period in compound interest problems?
5. How would the calculations change if interest were compounded semi-annually instead of annually?

Tip: Always carefully differentiate between simple and compound interest in problems, as they use different formulas and yield different results.