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 compound interest formula and the information provided in the question.

### Step 1: Understand the Compound Interest Formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \( A \) = Amount after time \( n \)
- \( P \) = Principal amount (initial sum of money)
- \( r \) = Rate of interest per annum
- \( n \) = Time in years

### Step 2: Set Up the Equation for the First Scenario
According to the question, the amount becomes three times the principal in 8 years. Thus, we can write:
\[ A = 3P \]
Substituting into the formula, we have:
\[ 3P = P \left(1 + \frac{r}{100}\right)^8 \]

### Step 3: Simplify the Equation
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = \left(1 + \frac{r}{100}\right)^8 \]
Let’s denote this as Equation (1).

### Step 4: Set Up the Equation for the Second Scenario
Now, we need to find out how many years it will take for the amount to become twenty-seven times the principal. Thus, we can write:
\[ A = 27P \]
Substituting into the formula, we have:
\[ 27P = P \left(1 + \frac{r}{100}\right)^n \]

### Step 5: Simplify the Second Equation
Dividing both sides by \( P \):
\[ 27 = \left(1 + \frac{r}{100}\right)^n \]
Let’s denote this as Equation (2).

### Step 6: Relate the Two Equations
From Equation (1), we have:
\[ \left(1 + \frac{r}{100}\right)^8 = 3 \]
From Equation (2), we have:
\[ \left(1 + \frac{r}{100}\right)^n = 27 \]
Notice that \( 27 = 3^3 \). Therefore, we can express Equation (2) as:
\[ \left(1 + \frac{r}{100}\right)^n = \left(\left(1 + \frac{r}{100}\right)^8\right)^3 \]
This implies:
\[ \left(1 + \frac{r}{100}\right)^n = \left(3\right)^3 \]

### Step 7: Set the Exponents Equal
Since the bases are the same, we can set the exponents equal:
\[ n = 8 \times 3 \]

### Step 8: Calculate \( n \)
Calculating this gives:
\[ n = 24 \]

### Final Answer
Thus, the money will become twenty-seven times of itself in **24 years**.

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