A certain sum becomes 3 times in 15 years at a certain rate of Interest. With another rate of interest becomes 4 times in 24 year. Find the different between rate of Interest.

Options

A. A. 1
B. B. `(5)/(6)`
C. C. `(2)/(5)`
D. D. `(9)/(7)`

Answer

To solve the problem step by step, we will use the concept of Simple Interest (SI) and the relationship between Principal (P), Rate of Interest (R), Time (T), and Amount (A).

### Step 1: Understand the problem
We are given two scenarios where a certain sum of money becomes three times and four times its value over different periods at different rates of interest. We need to find the difference between these rates of interest.

### Step 2: Define the variables
Let the principal amount be \( P \).

### Step 3: Calculate the amount and simple interest for the first case
In the first case, the amount becomes three times the principal in 15 years:
- Amount \( A_1 = 3P \)
- Simple Interest \( SI_1 = A_1 - P = 3P - P = 2P \)

### Step 4: Use the Simple Interest formula for the first case
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
For the first case:
\[ 2P = \frac{P \times R_1 \times 15}{100} \]
Cancelling \( P \) from both sides (assuming \( P \neq 0 \)):
\[ 2 = \frac{R_1 \times 15}{100} \]
Multiplying both sides by 100:
\[ 200 = R_1 \times 15 \]
Dividing both sides by 15:
\[ R_1 = \frac{200}{15} = \frac{40}{3} \text{ (Rate of Interest for the first case)} \]

### Step 5: Calculate the amount and simple interest for the second case
In the second case, the amount becomes four times the principal in 24 years:
- Amount \( A_2 = 4P \)
- Simple Interest \( SI_2 = A_2 - P = 4P - P = 3P \)

### Step 6: Use the Simple Interest formula for the second case
For the second case:
\[ 3P = \frac{P \times R_2 \times 24}{100} \]
Cancelling \( P \) from both sides:
\[ 3 = \frac{R_2 \times 24}{100} \]
Multiplying both sides by 100:
\[ 300 = R_2 \times 24 \]
Dividing both sides by 24:
\[ R_2 = \frac{300}{24} = \frac{75}{6} \text{ (Rate of Interest for the second case)} \]

### Step 7: Find the difference between the rates of interest
Now we need to find the difference between \( R_1 \) and \( R_2 \):
\[ \text{Difference} = R_1 - R_2 = \frac{40}{3} - \frac{75}{6} \]
To subtract these fractions, we need a common denominator. The common denominator of 3 and 6 is 6:
\[ R_1 = \frac{40}{3} = \frac{80}{6} \]
Now we can subtract:
\[ \text{Difference} = \frac{80}{6} - \frac{75}{6} = \frac{80 - 75}{6} = \frac{5}{6} \]

### Final Answer
The difference between the rates of interest is \( \frac{5}{6} \).

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