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The difference between compound interest (compounded annually) and simple interest for 3 years at the rate of 18% per annum is ₹25,758. What is the principal lent?
₹3,50,000
₹3,00,000
₹2,50,000
₹4,00,000
₹2,50,000
Use the 3-year difference formula directly: D=P(r/100)2(3+r/100). Calculate the percentage factor (0.18)2×3.18=0.0324×3.18≈0.103032. Dividing 25758 by this value yields the principal.
Difference between Compound Interest and Simple Interest (CI - SI) = ₹25,758, Time (T) = 3 years, Rate (R) = 18% per annum.
D3=P×(100R)2×(100300+R)
Use the 3-year difference formula directly: D=P(r/100)2(3+r/100). Calculate the percentage factor (0.18)2×3.18=0.0324×3.18≈0.103032. Dividing 25758 by this value yields the principal.
Many students mistake the 3-year formula for the 2-year formula, which is simply D=P(R/100)2, leading to an incorrect result.
Define the formula for 3 years
The difference between CI and SI for 3 years is given by the relation involving principal P and rate R.
D3=P[(1+100R)3−1−1003R]=P(100R)2(3+100R)
Substitute the given values
Plug R=18 and D3=25758 into the formula.
25758=P(10018)2×(3+10018)
Solve for P
Simplify the expression to find the value of P. 25758=P×0.0324×3.18.
P=0.0324×3.1825758=0.10303225758=250000
C is correct because the calculation of the principal yields exactly ₹2,50,000 using the difference formula for 3 years.
This formula is a specific application of the binomial expansion of (1+r)n. Understanding the derivation of this difference formula saves significant time in exam-hall calculations compared to calculating CI and SI separately.
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