If a sum of money at compound interest amounts to thrice itself in 3 years then in how many years will it be 9 times itself?

Answer

To solve the problem step by step, we will use the concept of compound interest.

### Step 1: Understand the given information
We know that a sum of money (let's call it P) amounts to thrice itself in 3 years. This means:
\[ A = 3P \]
where A is the amount after 3 years.

### Step 2: Set up the compound interest formula
The formula for the amount A in compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]
where:
- A is the amount after time t,
- P is the principal,
- R is the rate of interest,
- t is the time in years.

### Step 3: Substitute the known values into the formula
For the first scenario (3 years), we substitute A and t:
\[ 3P = P \left(1 + \frac{R}{100}\right)^3 \]

### Step 4: Simplify the equation
We can cancel P from both sides (assuming P ≠ 0):
\[ 3 = \left(1 + \frac{R}{100}\right)^3 \]

### Step 5: Take the cube root of both sides
To solve for \( 1 + \frac{R}{100} \), we take the cube root:
\[ 1 + \frac{R}{100} = 3^{1/3} \]

### Step 6: Express \( R \) in terms of \( 3^{1/3} \)
Now we can isolate R:
\[ \frac{R}{100} = 3^{1/3} - 1 \]
\[ R = 100(3^{1/3} - 1) \]

### Step 7: Set up the equation for the second scenario
Now we need to find out how long it will take for the amount to become 9 times the principal:
\[ A = 9P \]
Using the compound interest formula again:
\[ 9P = P \left(1 + \frac{R}{100}\right)^t \]

### Step 8: Simplify the equation again
Cancel P from both sides:
\[ 9 = \left(1 + \frac{R}{100}\right)^t \]

### Step 9: Substitute \( 1 + \frac{R}{100} \)
From Step 5, we know:
\[ 1 + \frac{R}{100} = 3^{1/3} \]
Substituting this into the equation gives:
\[ 9 = (3^{1/3})^t \]

### Step 10: Rewrite 9 as a power of 3
We know that:
\[ 9 = 3^2 \]
Thus, we can rewrite the equation as:
\[ 3^2 = (3^{1/3})^t \]

### Step 11: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ 2 = \frac{t}{3} \]

### Step 12: Solve for \( t \)
Multiplying both sides by 3 gives:
\[ t = 6 \]

### Conclusion
Thus, it will take 6 years for the sum of money to become 9 times itself.

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