Question

The difference between the CI and the SI on a sum of money lent for 2 years at 20% interest per annum is 80. The sum is:

• A. ₹2,000
• B. ₹1,200
• C. ₹1,500
• D. ₹1,000

Hint:

Compound interest and simple interest match exactly in the first year of any loan and only start to pull apart from the second year onward, because compound interest is calculated on the amount already grown by then, not on the original sum. Think about what extra amount that one difference in method actually adds over a 2-year period.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Compound Interest (CI) is interest calculated on the principal and the accumulated interest of previous periods. Simple Interest (SI) is calculated only on the principal. For the first year, CI and SI are the same, but they diverge from the second year onwards.

Step 2: Key Formula or Approach:
For a period of 2 years, the difference ($D$) between CI and SI is given by the formula: \[ D = P \left( \frac{R}{100} \right)^2 \] Where: $P$ = Principal (Sum), $R$ = Rate of interest per annum.

Step 3: Detailed Explanation:
Given: Difference ($D$) = 80 Rate ($R$) = 20% Time ($T$) = 2 years Substituting the values into the formula: \[ 80 = P \left( \frac{20}{100} \right)^2 \] \[ 80 = P \left( \frac{1}{5} \right)^2 \] \[ 80 = P \left( \frac{1}{25} \right) \] \[ P = 80 \times 25 \] \[ P = 2000 \]

Step 4: Final Answer:
The sum of money (Principal) is ₹2,000.

Answer 2

Concept:

• Compound interest and simple interest are identical for the first year of any loan - they only start to differ from the second year onward, because compound interest calculates interest on the accumulated amount from the previous year, rather than on the original sum alone.
• Assuming a convenient round principal, working out the actual rupee difference for that assumed value, and then scaling by direct proportion is often faster than recalling a memorised difference formula.

Step 1: Assume a convenient principal value.
Let the assumed principal be $P_a = 100$. Since both compound interest and simple interest scale directly with the principal, the ratio between the difference and the principal stays the same no matter what value is assumed here.

Step 2: Compute simple interest for 2 years on this assumed principal.
$SI = \dfrac{P_a \times R \times T}{100} = \dfrac{100 \times 20 \times 2}{100} = 40$. Each of the two years contributes $20$ equally.

Step 3: Compute compound interest for 2 years on the same assumed principal, year by year.
Year 1 interest $= 20\%$ of $100 = 20$, taking the amount to $120$. Year 2 interest $= 20\%$ of $120 = 24$, since compound interest is calculated on the new amount, not the original sum. Total $CI = 20 + 24 = 44$.

Step 4: Find the difference for the assumed value, then scale up to the real difference given in the question.
For $P_a = 100$, the difference is $44 - 40 = 4$. Since this difference is directly proportional to the principal, set up: $4$ corresponds to $100$, so $80$ corresponds to $100 \times \dfrac{80}{4} = 2000$.

Final Answer: ₹2,000