Question

A certain sum of money becomes 625/256 times of itself in 1 year. Find the rate of interest per annum if interest is compounded quarterly?

• A. 5
• B. 25
• C. 50
• D. 100

Answer 1

The Correct Option is D

The formula for compound interest is given by: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount after \( t \) years - \( P \) is the principal amount (initial sum of money) - \( r \) is the rate of interest per annum - \( n \) is the number of times interest is compounded per year - \( t \) is the time in years In this case, we are given that the amount becomes \( \frac{625}{256} \) times the initial amount (\( A = \frac{625}{256}P \)) in 1 year, and the interest is compounded quarterly (\( n = 4 \)). Substituting the given values, we have: \[ \frac{625}{256}P = P \left(1 + \frac{r}{4}\right)^{4 \cdot 1} \] Simplifying: \[ \frac{625}{256} = \left(1 + \frac{r}{4}\right)^4 \] Taking the fourth root of both sides: \[ \sqrt[4]{\frac{625}{256}} = 1 + \frac{r}{4} \] Now, solving for \( r \): \[ \frac{5}{4} = 1 + \frac{r}{4} \] Subtracting 1 from both sides: \[ \frac{1}{4} = \frac{r}{4} \] Multiplying both sides by 4: \[ 1 = r \] So, the rate of interest per annum is 100%. Therefore, the answer is 100%, which corresponds to the option: 100. 

Answer 2

Given that the amount becomes \( \left( \frac{625}{256} \right)x \) after 1 year, compounded quarterly, we are to find the annual rate of interest.

Step 1: Let Principal \( P = x \)

Then, Amount \( A = \left( \frac{625}{256} \right)x \)

Step 2: Time and Rate Details

• Time \( T = 1 \) year \( = 4 \) quarters
• Let annual rate \( = r\% \Rightarrow \) Quarterly rate \( = \frac{r}{4}\% \)

Step 3: Compound Interest Formula

The compound interest formula is: \[ A = P \left(1 + \frac{r}{100n} \right)^{nt} \] Here, since compounding is quarterly: \[ A = P \left(1 + \frac{r}{400} \right)^4 \]

Step 4: Plug in Given Values

\[ \left( \frac{625}{256} \right)x = x \left(1 + \frac{r}{400} \right)^4 \] Cancel \( x \) from both sides: \[ \frac{625}{256} = \left(1 + \frac{r}{400} \right)^4 \]

Step 5: Recognize Perfect Powers

\[ \left( \frac{625}{256} \right) = \left( \frac{5}{4} \right)^4 \Rightarrow \left( \frac{5}{4} \right)^4 = \left(1 + \frac{r}{400} \right)^4 \] Taking fourth roots on both sides: \[ \frac{5}{4} = 1 + \frac{r}{400} \]

Step 6: Solve for \( r \)

\[ \frac{5}{4} - 1 = \frac{r}{400} \Rightarrow \frac{1}{4} = \frac{r}{400} \] \[ r = \frac{400}{4} = 100 \]

Final Answer:

\[ \boxed{r = 100\% \text{ per annum}} \]

Correct Option: (D) 100