1. Problem 1 (Finding Rate): Given a principal of $1000$, a simple interest of $150$, and a time period of $3$ years, find the rate of interest. Step 1: Write the formula for simple interest: $$SI = \frac{P \times R \times T}{100}$$ Step 2: Substitute the given values: $$150 = \frac{1000 \times R \times 3}{100}$$ Step 3: Solve for $R$: $$150 = 30R \implies R = \frac{150}{30} = 5$$ Answer: Rate of interest is $5\%$ per annum. 2. Problem 2 (Finding Time): Given a principal of $2000$, a simple interest of $400$, and a rate of interest of $8\%$ per annum, find the time period. Step 1: Write the formula for simple interest: $$SI = \frac{P \times R \times T}{100}$$ Step 2: Substitute the given values: $$400 = \frac{2000 \times 8 \times T}{100}$$ Step 3: Simplify and solve for $T$: $$400 = 160T \implies T = \frac{400}{160} = 2.5$$ Answer: Time period is $2.5$ years. 3. Problem 3 (Finding Principal): Given a simple interest of $180$, rate of interest $6\%$ per annum, and time period $5$ years, find the principal amount. Step 1: Write the formula for simple interest: $$SI = \frac{P \times R \times T}{100}$$ Step 2: Substitute the given values: $$180 = \frac{P \times 6 \times 5}{100}$$ Step 3: Simplify and solve for $P$: $$180 = \frac{30P}{100} = 0.3P \implies P = \frac{180}{0.3} = 600$$ Answer: Principal amount is $600$.