A sum of money becomes 5 times of itself in 5 years at simple interest. In what time it will becomes 7 times of itself.

Answer

To solve the problem step by step, we will break down the information given and apply the formula for simple interest.

### Step 1: Understand the Problem
We know that a sum of money (let's call it P) becomes 5 times itself in 5 years at simple interest. This means:
- Amount (A) after 5 years = 5P
- Principal (P) = P
- Time (T) = 5 years

### Step 2: Set Up the Simple Interest Formula
The formula for the amount in simple interest is:
\[ A = P + \text{SI} \]
Where SI (Simple Interest) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]

### Step 3: Calculate the Simple Interest
From the information given:
\[ 5P = P + \frac{P \times R \times 5}{100} \]

### Step 4: Rearranging the Equation
Subtract P from both sides:
\[ 5P - P = \frac{P \times R \times 5}{100} \]
This simplifies to:
\[ 4P = \frac{P \times R \times 5}{100} \]

### Step 5: Cancel P from Both Sides
Assuming P is not zero, we can divide both sides by P:
\[ 4 = \frac{R \times 5}{100} \]

### Step 6: Solve for R (Rate of Interest)
Multiply both sides by 100:
\[ 400 = R \times 5 \]
Now divide by 5:
\[ R = \frac{400}{5} = 80\% \]

### Step 7: Find the Time to Become 7 Times Itself
Now we need to find out how long it will take for the amount to become 7 times itself:
- Amount (A) = 7P
- Principal (P) = P
- Rate (R) = 80%
- We need to find Time (T).

Using the simple interest formula again:
\[ 7P = P + \frac{P \times 80 \times T}{100} \]

### Step 8: Rearranging the Equation Again
Subtract P from both sides:
\[ 7P - P = \frac{P \times 80 \times T}{100} \]
This simplifies to:
\[ 6P = \frac{P \times 80 \times T}{100} \]

### Step 9: Cancel P from Both Sides
Again, assuming P is not zero:
\[ 6 = \frac{80 \times T}{100} \]

### Step 10: Solve for T
Multiply both sides by 100:
\[ 600 = 80T \]
Now divide by 80:
\[ T = \frac{600}{80} = 7.5 \text{ years} \]

### Final Answer
It will take 7.5 years for the sum of money to become 7 times itself.

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