If a sum of money becomes 4000 in 2 year and 5500 in 4 year 6 month at the same rate of simple interest per annum. Then the rate of simple interest is :

Options

A. `21(3)/(7)%`
B. `21(2)/(7)%`
C. `21(1)/(7)%`
D. `21(5)/(7)%`

Answer

To solve the problem step-by-step, we will follow the information given in the question and the calculations discussed in the video transcript.

### Step 1: Understand the problem
We have two amounts of money at two different times:
- Amount after 2 years = 4000
- Amount after 4 years 6 months = 5500

### Step 2: Calculate the difference in time and amount
The time difference between the two amounts is:
- From 2 years to 4 years 6 months = 4.5 years - 2 years = 2.5 years

The difference in the amounts is:
- 5500 - 4000 = 1500

### Step 3: Calculate the simple interest for the time difference
The simple interest for the 2.5 years is 1500.

### Step 4: Calculate the simple interest for 1 year
Using the unitary method, we can find the simple interest for 1 year:
- Simple Interest for 1 year = Total Simple Interest for 2.5 years / 2.5
- Simple Interest for 1 year = 1500 / 2.5 = 600

### Step 5: Calculate the total simple interest for 2 years
Now, we can calculate the simple interest for 2 years:
- Simple Interest for 2 years = 2 * Simple Interest for 1 year = 2 * 600 = 1200

### Step 6: Find the principal amount
We know that the amount after 2 years is 4000. Thus, we can find the principal amount:
- Principal = Amount after 2 years - Simple Interest for 2 years
- Principal = 4000 - 1200 = 2800

### Step 7: Use the simple interest formula to find the rate
The formula for simple interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest
- P = Principal
- R = Rate of interest
- T = Time in years

We can rearrange this formula to find the rate \( R \):
\[ R = \frac{SI \times 100}{P \times T} \]

Substituting the known values:
- SI = 1200
- P = 2800
- T = 2 years

Now, substituting these values into the formula:
\[ R = \frac{1200 \times 100}{2800 \times 2} \]
\[ R = \frac{120000}{5600} \]
\[ R = 21.42857 \]

### Step 8: Convert the rate to a percentage
To express the rate as a percentage, we can simplify:
\[ R \approx 21.43\% \]

### Final Answer
The rate of simple interest is approximately **21.43%**.

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