A sum of money lent out at C.I. at a certain rate per annum becomes three times of itself in 8 years. Find in how many years will the money become twenty-seven times of itself at the same rate of interest p.a.

Answer

To solve the problem step by step, we will use the formula for compound interest and the information provided in the question.

### Step 1: Understand the Problem
We need to find out how many years it will take for a sum of money to become 27 times itself at the same rate of interest per annum, given that it becomes 3 times itself in 8 years.

### Step 2: Set Up the Variables
Let:
- \( P \) = Principal amount (initial sum of money)
- \( R \) = Rate of interest per annum
- \( A \) = Amount after time \( t \)

### Step 3: Use the Compound Interest Formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]

### Step 4: Apply the First Condition
According to the problem, the amount becomes 3 times itself in 8 years:
\[ A = 3P \]
Substituting this into the formula:
\[ 3P = P \left(1 + \frac{R}{100}\right)^8 \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = \left(1 + \frac{R}{100}\right)^8 \]

### Step 5: Simplify the Equation
This gives us our first equation:
\[ \left(1 + \frac{R}{100}\right)^8 = 3 \]
Let’s call this Equation (1).

### Step 6: Apply the Second Condition
Now, we need to find out how many years it will take for the amount to become 27 times itself:
\[ A = 27P \]
Substituting this into the formula:
\[ 27P = P \left(1 + \frac{R}{100}\right)^n \]
Again, dividing both sides by \( P \):
\[ 27 = \left(1 + \frac{R}{100}\right)^n \]

### Step 7: Rewrite 27 in Terms of 3
Since \( 27 = 3^3 \), we can rewrite the equation:
\[ 3^3 = \left(1 + \frac{R}{100}\right)^n \]

### Step 8: Relate the Two Equations
From Equation (1), we know:
\[ \left(1 + \frac{R}{100}\right)^8 = 3 \]
Thus, we can substitute \( 3 \) from Equation (1) into our new equation:
\[ (3)^3 = \left(1 + \frac{R}{100}\right)^n \]
This implies:
\[ 3^3 = \left(\left(1 + \frac{R}{100}\right)^8\right)^{\frac{n}{8}} \]

### Step 9: Use the Power of a Power Rule
Using the property of exponents:
\[ 3^3 = 3^{\frac{n}{8}} \]
Since the bases are the same, we can equate the exponents:
\[ 3 = \frac{n}{8} \]

### Step 10: Solve for \( n \)
Multiplying both sides by 8:
\[ n = 24 \]

### Conclusion
Thus, it will take **24 years** for the money to become 27 times itself at the same rate of interest per annum.

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