A sum becomes two times in 5 yr at a certain rate of interest. Find the time in which the same amount will be 8 times at the same rate of interest.

Answer

To solve the problem step by step, we will use the concept of Simple Interest and the relationship between Principal, Interest, and Time.

### Step 1: Understand the given information
We know that a certain sum of money (let's call it Principal, P) becomes two times in 5 years. This means that the amount (A) after 5 years is:
\[ A = 2P \]

### Step 2: Calculate the interest earned in 5 years
The interest (I) earned in 5 years can be calculated as:
\[ I = A - P \]
Substituting the value of A:
\[ I = 2P - P = P \]

### Step 3: Find the rate of interest
Using the formula for Simple Interest:
\[ I = P \times r \times t \]
Where:
- I = Interest
- P = Principal
- r = Rate of interest (in decimal)
- t = Time in years

We can substitute the values we have for the first scenario:
\[ P = P \times r \times 5 \]
This simplifies to:
\[ I = P = P \times r \times 5 \]
Dividing both sides by P (assuming P is not zero):
\[ 1 = r \times 5 \]
Thus, we find:
\[ r = \frac{1}{5} = 0.2 \text{ or } 20\% \]

### Step 4: Determine the time for the amount to become 8 times
Now we want to find the time (t) when the amount becomes 8 times the principal:
\[ A = 8P \]
The interest earned in this case will be:
\[ I = A - P = 8P - P = 7P \]

Using the Simple Interest formula again:
\[ I = P \times r \times t \]
Substituting the values we have:
\[ 7P = P \times 0.2 \times t \]
Dividing both sides by P:
\[ 7 = 0.2 \times t \]
Now, solve for t:
\[ t = \frac{7}{0.2} = 35 \text{ years} \]

### Final Answer
The time in which the same amount will be 8 times at the same rate of interest is **35 years**.

---