If 2.5252525…….. `=(p)/(q)` (in the lowest form) then what is the value of `(q)/(p)?`

Options

A. 0.4
B. 0.42525
C. 0.0396
D. 0.396

Answer

To solve the problem where \( 2.5252525\ldots = \frac{p}{q} \) and we need to find the value of \( \frac{q}{p} \), we can follow these steps:

### Step 1: Express the repeating decimal as a fraction
Let \( x = 2.5252525\ldots \).

### Step 2: Separate the whole number and the repeating part
We can separate the whole number from the decimal:
\[ x = 2 + 0.5252525\ldots \]

### Step 3: Let the repeating part be another variable
Let \( y = 0.5252525\ldots \).

### Step 4: Multiply \( y \) by 100 to shift the decimal point
\[ 100y = 52.5252525\ldots \]

### Step 5: Set up the equation
Now we can set up the equation:
\[ 100y = 52 + y \]

### Step 6: Solve for \( y \)
Rearranging gives:
\[ 100y - y = 52 \]
\[ 99y = 52 \]
\[ y = \frac{52}{99} \]

### Step 7: Substitute back to find \( x \)
Now substitute \( y \) back into the expression for \( x \):
\[ x = 2 + \frac{52}{99} \]

### Step 8: Convert the whole number to a fraction
Convert 2 into a fraction with a denominator of 99:
\[ x = \frac{198}{99} + \frac{52}{99} = \frac{198 + 52}{99} = \frac{250}{99} \]

### Step 9: Identify \( p \) and \( q \)
From the fraction \( \frac{250}{99} \), we have \( p = 250 \) and \( q = 99 \).

### Step 10: Find \( \frac{q}{p} \)
Now we need to find \( \frac{q}{p} \):
\[ \frac{q}{p} = \frac{99}{250} \]

### Step 11: Calculate the decimal value
To find the decimal value:
\[ \frac{99}{250} = 0.396 \]

### Final Answer
Thus, the value of \( \frac{q}{p} \) is \( 0.396 \).

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