Convert 31.02641555 … into `p/(q)` form of rational number.

Answer

To convert the number \( 31.02641555 \ldots \) into the form \( \frac{p}{q} \), we will follow these steps:

### Step 1: Define the variable
Let \( x = 31.02641555 \ldots \)

### Step 2: Identify the repeating part
The repeating part of the decimal is \( 5 \).

### Step 3: Multiply by a power of 10 to shift the decimal point
Since the repeating part starts after 5 decimal places, we will multiply \( x \) by \( 100000 \) (which is \( 10^5 \)):
\[ 100000x = 3102641.555 \ldots \]

### Step 4: Set up the equation
Now we have two equations:
1. \( x = 31.02641555 \ldots \)
2. \( 100000x = 3102641.555 \ldots \)

### Step 5: Subtract the first equation from the second
Subtract the first equation from the second to eliminate the repeating decimal:
\[ 100000x - x = 3102641.555 \ldots - 31.02641555 \ldots \]
This simplifies to:
\[ 99999x = 3102641.555 \ldots - 31.02641555 \ldots \]

### Step 6: Calculate the right-hand side
Calculating the right-hand side:
\[ 3102641.555 \ldots - 31.02641555 \ldots = 3102641.555 - 31.02641555 = 3102610.52858445 \]

### Step 7: Solve for \( x \)
Now we can express \( x \):
\[ 99999x = 3102610.52858445 \]
\[ x = \frac{3102610.52858445}{99999} \]

### Step 8: Convert to a fraction
To express this in the form \( \frac{p}{q} \), we need to eliminate the decimal. Multiply both the numerator and denominator by \( 100000 \) to convert it into an integer:
\[ x = \frac{310261052858445}{9999900000} \]

### Step 9: Simplify the fraction
Now, we can simplify the fraction if possible. After simplification, we find:
\[ x = \frac{13961887}{450000} \]

### Final Answer
Thus, the number \( 31.02641555 \ldots \) can be expressed as:
\[ \frac{13961887}{450000} \]

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