The least number of complete years in which, a sum of money put at `20%` CI will be more than doubled is :

Options

A. 4
B. 5
C. 6
D. 8

Answer

To solve the problem of finding the least number of complete years in which a sum of money put at 20% compound interest (CI) will be more than doubled, we can follow these steps:

### Step 1: Understand the Problem
We need to determine how many years it will take for an initial amount (let's call it P) to become more than double (i.e., more than 2P) when compounded annually at a rate of 20%.

**Hint:** Remember that doubling means we want the final amount to be greater than 2 times the principal.

### Step 2: Use the Compound Interest Formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial sum of money).
- \( r \) is the annual interest rate (in percentage).
- \( n \) is the number of years the money is invested or borrowed.

**Hint:** You will substitute \( r = 20 \) and \( A > 2P \).

### Step 3: Set Up the Inequality
We need:
\[ P \left(1 + \frac{20}{100}\right)^n > 2P \]
This simplifies to:
\[ \left(1.2\right)^n > 2 \]

**Hint:** You can divide both sides by P since P is positive.

### Step 4: Solve for n
To solve for \( n \), we can take the logarithm of both sides:
\[ n \log(1.2) > \log(2) \]
Thus,
\[ n > \frac{\log(2)}{\log(1.2)} \]

**Hint:** Use a calculator to find the values of \( \log(2) \) and \( \log(1.2) \).

### Step 5: Calculate the Values
Using a calculator:
- \( \log(2) \approx 0.3010 \)
- \( \log(1.2) \approx 0.0792 \)

Now substitute these values into the inequality:
\[ n > \frac{0.3010}{0.0792} \approx 3.8 \]

**Hint:** Since we need the least number of complete years, round up to the next whole number.

### Step 6: Conclusion
Since \( n \) must be a complete year, we round 3.8 up to 4. Therefore, the least number of complete years in which the sum of money will be more than doubled at 20% CI is **4 years**.

**Final Answer:** 4 years