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 \( 2.5252525... = \frac{p}{q} \) in its lowest form, and we need to find the value of \( \frac{q}{p} \), we will follow these steps:

### Step 1: Identify the repeating decimal
The number \( 2.5252525... \) can be expressed as \( 2.52\overline{52} \), where \( 52 \) is the repeating part.

### Step 2: Separate the whole number and the decimal part
We can separate the whole number from the decimal part:
\[ 2.52\overline{52} = 2 + 0.52\overline{52} \]

### Step 3: Convert the repeating decimal to a fraction
To convert \( 0.52\overline{52} \) to a fraction, we can use the formula for converting repeating decimals:
\[ x = 0.52\overline{52} \]
Multiply both sides by \( 100 \) (to shift the decimal point two places to the right):
\[ 100x = 52.52\overline{52} \]
Now, subtract the original \( x \) from this equation:
\[ 100x - x = 52.52\overline{52} - 0.52\overline{52} \]
This simplifies to:
\[ 99x = 52 \]
Thus, we find:
\[ x = \frac{52}{99} \]

### Step 4: Combine the whole number and the fraction
Now, we can combine the whole number and the fraction:
\[ 2.52\overline{52} = 2 + \frac{52}{99} = \frac{2 \times 99 + 52}{99} = \frac{198 + 52}{99} = \frac{250}{99} \]

### Step 5: Identify \( p \) and \( q \)
From the fraction \( \frac{250}{99} \), we identify:
- \( p = 250 \)
- \( q = 99 \)

### Step 6: Calculate \( \frac{q}{p} \)
Now we can find \( \frac{q}{p} \):
\[ \frac{q}{p} = \frac{99}{250} \]

### Step 7: Simplify if necessary
The fraction \( \frac{99}{250} \) is already in its lowest form.

### Final Answer
Thus, the value of \( \frac{q}{p} \) is:
\[ \frac{q}{p} = \frac{99}{250} \]