The following are the steps involved in converting `1.bar(52)` into p/q form where `q!=0`. Arrange them in sequential order from the first to last.
A. `:.10x=15.2222`…….2
B. Let `x=1.bar(52)=1.5 2222`……..1
c. `9x=13.7impliesx=13.7/9=137/90`
D. As periodicity is 1 multiply the Eq (1) with 10.
E. Subtract Eq. 1 from the Eq. 2

Options

A. BDAEC
B. DBAEC
C. DBCEA
D. BDCEA

Answer

To convert \( 1.\overline{52} \) into the form \( \frac{p}{q} \), we can follow these steps in the correct sequential order:

### Step-by-Step Solution

1. **Let \( x = 1.\overline{52} \)**
This means we define \( x \) as the repeating decimal.
**Equation 1:** \( x = 1.525252... \)
(This corresponds to step B)

2. **Multiply by 10**
Since the periodicity of the repeating part (52) is 2, we multiply \( x \) by 100 instead of 10. However, since the question states periodicity is 1, we will multiply by 10 for the sake of this sequence.
**Equation 2:** \( 10x = 15.252525... \)
(This corresponds to step D)

3. **Subtract Equation 1 from Equation 2**
We subtract the first equation from the second to eliminate the repeating part.
\( 10x - x = 15.252525... - 1.525252... \)
This simplifies to:
\( 9x = 13.7 \)
(This corresponds to step E)

4. **Solve for \( x \)**
Now, we can solve for \( x \) by dividing both sides by 9.
\( x = \frac{13.7}{9} \)
To express it in fraction form, we convert 13.7 to a fraction:
\( 13.7 = \frac{137}{10} \)
Therefore,
\( x = \frac{137}{10 \times 9} = \frac{137}{90} \)
(This corresponds to step C)

### Final Order of Steps
The correct order of steps is:
- B: Let \( x = 1.\overline{52} = 1.525252... \)
- D: Multiply the equation by 10 (or 100 for periodicity of 2).
- E: Subtract Equation 1 from Equation 2.
- C: Solve for \( x \).