Shortcut Trick
Effective rate for 2 years (compounded annually at 20%) = 20 + 20 + (20 × 20)/100 = 44%.
Effective rate for 2 years (compounded half-yearly at 10% per cycle) = 46.41%.
Difference in interest rate percentage = 46.41% − 44% = 2.41%.
Value of x = 2.41% of ₹16,000 = 2.41 × 160 = ₹385.60.
∴ The correct answer is ₹385.60.
Alternate Method
Given:
Principal (P) = ₹16,000
Time (T) = 2 years
Rate (R) = 20% per annum
Formula Used:
Amount (A) = P(1 + r/100)n
Compound Interest (CI) = A − P
Case 1: Compounded Annually
⇒ Rate = 20%, Time = 2 years
⇒ Amount = 16000(1 + 20/100)2 = 16000(1.2)2
⇒ Amount = 16000 × 1.44 = ₹23,040
⇒ CI1 = 23040 − 16000 = ₹7,040
Case 2: Compounded Half-yearly
⇒ New Rate = 20/2 = 10% per half-year
⇒ New Time = 2 × 2 = 4 half-years
⇒ Amount = 16000(1 + 10/100)4 = 16000(1.1)4
⇒ Amount = 16000 × 1.4641 = ₹23,425.60
⇒ CI2 = 23425.60 − 16000 = ₹7,425.60
Difference (x):
⇒ x = CI2 − CI1
⇒ x = 7425.60 − 7040.00 = ₹385.60
∴ The correct answer is ₹385.60.
Additional Information
Half-yearly Compounding Rule
When interest is compounded half-yearly, the annual rate is divided by 2 and the number of years is multiplied by 2.
Effective Rate of Interest
For a rate r for 2 periods, the effective rate is [r + r + (r × r)/100] %.
Compound Interest vs Simple Interest
For 2 years, the difference between CI and SI is given by P(R/100)2.