At what rate percent by simple interest, will a sum of money double itself in 5 years 4 months ?

Answer

To find the rate percent by simple interest at which a sum of money will double itself in 5 years and 4 months, we can follow these steps:

### Step 1: Understand the Problem
We need to determine the rate of simple interest at which a principal amount (P) doubles in a specific time period.

### Step 2: Define the Variables
Let the principal amount be \( P = x \).
Since the amount doubles, the final amount \( A = 2x \).

### Step 3: Calculate the Simple Interest (SI)
The simple interest earned when the amount doubles is given by:
\[ SI = A - P = 2x - x = x \]

### Step 4: Convert Time into Years
The time given is 5 years and 4 months. We need to convert this into years:
\[ 5 \text{ years} + 4 \text{ months} = 5 + \frac{4}{12} = 5 + \frac{1}{3} = \frac{16}{3} \text{ years} \]

### Step 5: Use the Simple Interest Formula
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) is the simple interest,
- \( P \) is the principal,
- \( R \) is the rate of interest,
- \( T \) is the time in years.

Substituting the known values into the formula:
\[ x = \frac{x \times R \times \frac{16}{3}}{100} \]

### Step 6: Simplify the Equation
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 1 = \frac{R \times \frac{16}{3}}{100} \]

### Step 7: Solve for R
Rearranging the equation to solve for \( R \):
\[ R = \frac{100 \times 1 \times 3}{16} = \frac{300}{16} = \frac{75}{4} \]

### Step 8: Convert to Percentage
Now, converting \( \frac{75}{4} \) into a mixed fraction:
\[ \frac{75}{4} = 18 \frac{3}{4} \]
Thus, the rate of interest is \( 18.75\% \).

### Final Answer
The rate percent by simple interest at which a sum of money will double itself in 5 years and 4 months is **18.75%**.

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