Step 1 : Understanding the Question
This question compares Compound Interest (CI) and Simple Interest (SI). While SI is calculated only on the principal, CI includes "interest on interest." For the first year, both interests are identical, but in the second year, CI becomes higher because interest is also charged on the amount earned in the first year. We are given the difference between these two values after two years and must find the original principal (the sum).
Step 2 : Key Formulas and approach
Instead of calculating CI and SI separately, which is time-consuming, we use a specialized "Difference Formula" for a 2-year period. This formula directly links the principal, the interest rate, and the difference.
Key Formula:
$\text{Difference (D)} = P \left( \frac{R}{100} \right)^2$
(Where $P$ is the principal, $R$ is the rate, and $D$ is the given difference).
Step 3 : Detailed Explanation
Identifying Given Values: The problem provides the Difference ($D$) as Rs80 and the Rate of interest ($R$) as 20%. The time period is 2 years, which matches our formula's requirement.
Substituting into the Formula: We write the equation as $80 = P \left( \frac{20}{100} \right)^2$.
Simplifying the Rate Fraction: The fraction $\frac{20}{100}$ simplifies to $\frac{1}{5}$. Our equation now becomes $80 = P \times \left( \frac{1}{5} \right)^2$.
Squaring the Fraction: Squaring $\frac{1}{5}$ gives $\frac{1}{25}$. So, $80 = \frac{P}{25}$.
Calculating the Principal: To find $P$, we multiply 80 by 25. $80 \times 25 = 2000$. Thus, the initial sum lent was Rs2,000.
Step 4 : Final Answer
The sum of money lent is Rs2,000, which corresponds to option (A).