Question

Given below are two statements :
Statement I: A savings account at Bank A pays 6.2% interest, compounded annually. Bank B's savings account pays 6% compounded semi-annually. Bank B is paying less total interest each year.
Statement II: A sum of money at a certain rate of compound interest doubles in 3 years. In 9 years, it will be P times original principal. Then P = 9.
In the light of the above statements, choose the correct answer from the options given below.

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is C

To evaluate the given statements, we will analyze each one step-by-step:

1. Statement I: A savings account at Bank A pays 6.2% interest, compounded annually. Bank B's savings account pays 6% compounded semi-annually. Bank B is paying less total interest each year.
   • We first need to calculate the effective annual rate (EAR) for each bank, as the compounding frequencies differ.
   • For Bank A, which compounds annually at 6.2%, the effective interest rate remains 6.2%.
   • For Bank B, the semi-annual compounding requires the following formula to determine the EAR:
     • The formula for EAR is: \(EAR = \left(1 + \frac{r}{n}\right)^n - 1\) where \(r\) is the nominal rate and \(n\) is the number of compounding periods.
     • Here, \(r = 6\%\) and \(n = 2\) (because of the semi-annual compounding).
     • Plugging in the values, we get:
   • Since Bank A offers 6.2% and Bank B effectively offers 6.09%, Bank B pays less interest, making Statement I true.
2. Statement II: A sum of money at a certain rate of compound interest doubles in 3 years. In 9 years, it will be P times the original principal. Then P = 9.
   • The formula for compound interest is given by: \(A = P \times (1 + r)^t\) where \(A\) is the amount of money, \(P\) is the principal, \(r\) is the rate, and \(t\) is the number of years.
   • Given that the principal doubles in 3 years, the relationship is:
   • Solving for \(1 + r\) gives:
   • After 9 years, the amount becomes:
   • Hence, the principal is multiplied by 8, not 9, making Statement II false. 

Based on these analyses, the correct answer is that Statement I is true but Statement II is false.