Simple Interest on a certain sum at a certain annual rate of interest is `(16)/(25)` of the sum. If the number representing rate per cent and time in years be equal. then the rate of interest is

Options

A. 0.08
B. `11 (1)/(2)%`
C. `12(1)/(2)%`
D. `12(1)/(4)%`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the relationship given in the problem
The problem states that the simple interest (SI) on a certain sum (let's denote it as P) at a certain annual rate of interest (r) is \( \frac{16}{25} \) of the sum (P).

### Step 2: Set up the equation
Let the principal amount (P) be represented as \( 25x \). Therefore, the simple interest (SI) can be expressed as:
\[ SI = \frac{16}{25} \times P = \frac{16}{25} \times 25x = 16x \]

### Step 3: Use the formula for Simple Interest
The formula for simple interest is given by:
\[ SI = \frac{P \times r \times t}{100} \]
In this case, since the rate (r) and time (t) are equal, we can denote both as \( r \). Thus, the formula becomes:
\[ SI = \frac{P \times r \times r}{100} = \frac{P \times r^2}{100} \]

### Step 4: Substitute the values into the formula
Substituting \( P = 25x \) and \( SI = 16x \) into the formula gives us:
\[ 16x = \frac{25x \times r^2}{100} \]

### Step 5: Simplify the equation
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 16 = \frac{25r^2}{100} \]
Multiplying both sides by 100 to eliminate the fraction:
\[ 1600 = 25r^2 \]

### Step 6: Solve for \( r^2 \)
Dividing both sides by 25:
\[ r^2 = \frac{1600}{25} = 64 \]

### Step 7: Find \( r \)
Taking the square root of both sides:
\[ r = \sqrt{64} = 8 \]

### Conclusion
Thus, the rate of interest is \( 8\% \).

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