To solve this problem, let's break it down step-by-step.

Step 1: Formula for Simple Interest (SI)

The formula for simple interest is:

$SI = \frac{P \times R \times T}{100}$

Where:

• $P$ = Principal amount
• $R$ = Rate of interest per annum
• $T$ = Time in years

Step 2: Formula for Compound Interest (CI)

The formula for compound interest is:

$CI = P \left(1 + \frac{R}{100}\right)^T - P$

Where:

• $P$ = Principal amount
• $R$ = Rate of interest per annum
• $T$ = Time in years

Step 3: Difference Between CI and SI

The problem states that the difference between CI and SI is Rs.180.5. So,

$CI - SI = 180.5$

Step 4: Plugging in the Variables

Let the sum invested (principal) be $P$, the rate $R = 19\%$, and the time $T = 2$ years.

1. Calculate $SI$: $SI = \frac{P \times 19 \times 2}{100} = \frac{38P}{100} = 0.38P$

2. Calculate $CI$: Using the compound interest formula: $CI = P \left(1 + \frac{19}{100}\right)^2 - P$ Simplify: $CI = P \left(1.19\right)^2 - P$ $CI = P \times 1.4161 - P = 1.4161P - P = 0.4161P$

3. Difference between $CI$ and $SI$: $CI - SI = 0.4161P - 0.38P = 0.0361P$

4. Set the difference equal to Rs.180.5: $0.0361P = 180.5$

Step 5: Solve for $P$

$P = \frac{180.5}{0.0361} = 5000$

Final Answer:

The sum invested is Rs. 5000.

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Would you like more details about the calculations? Here are some related questions to expand your understanding:

1. How does the time period affect the difference between simple and compound interest?
2. What happens if the rate of interest increases in the compound interest scheme?
3. How do we compute compound interest if the compounding frequency is quarterly instead of annually?
4. What is the relationship between principal and the difference in interest for longer time periods?
5. Can you derive the general formula for the difference between CI and SI?

Tip: Always check whether the compounding frequency matches the time period in compound interest problems.