The simple interest on a sum of money is 1/144 of the principal and the number of years is equal to the rate per cent per annum. What will be the rate per cent per annum?

Options

A. `(3)/(5)%`
B. `(5)/(6)%`
C. `(7)/(8)%`
D. `(1)/(6)%`

Answer

To solve the problem, we need to use the formula for Simple Interest (SI):

\[ SI = \frac{P \times R \times T}{100} \]

Where:
- \( SI \) = Simple Interest
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

From the question, we know:
1. The Simple Interest (SI) is \( \frac{1}{144} \) of the Principal (P).
2. The number of years (T) is equal to the rate of interest (R).

### Step 1: Set up the equations
Let \( P \) be the principal amount. Then, according to the problem:

\[ SI = \frac{1}{144} P \]

And since \( T = R \), we can denote both as \( R \).

### Step 2: Substitute SI in the formula
Substituting the value of SI into the Simple Interest formula gives us:

\[ \frac{1}{144} P = \frac{P \times R \times R}{100} \]

### Step 3: Simplify the equation
We can simplify this equation by multiplying both sides by 14400 (to eliminate the fraction):

\[ P = \frac{P \times R^2 \times 144}{100} \]

### Step 4: Cancel \( P \) from both sides
Assuming \( P \neq 0 \), we can cancel \( P \) from both sides:

\[ 1 = \frac{R^2 \times 144}{100} \]

### Step 5: Rearranging the equation
Now, rearranging the equation gives:

\[ R^2 \times 144 = 100 \]

### Step 6: Solve for \( R^2 \)
Dividing both sides by 144:

\[ R^2 = \frac{100}{144} \]

### Step 7: Simplifying \( R^2 \)
Simplifying \( \frac{100}{144} \):

\[ R^2 = \frac{25}{36} \]

### Step 8: Taking the square root
Taking the square root of both sides gives us:

\[ R = \sqrt{\frac{25}{36}} = \frac{5}{6} \]

### Step 9: Convert to percentage
To convert \( R \) into a percentage, we multiply by 100:

\[ R = \frac{5}{6} \times 100 = 83.33\% \]

### Final Answer
The rate of interest per annum is \( 83.33\% \).

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