Match the following:
`{:("Column-I"," Column-II"),( "(P) Rational form of " 0.bar(32) is ,(i)(14)/(55)),("(Q) Rational form of " 0.2 bar(54) is ,(ii) (11)/(45)),("(R) Rational form of" 0.bar(12) is ,(iii) (32)/(99)),("(S) Rational form of 0.2bar(4) is, (iv) (11)/(90)):}`

Options

A. `(P) rarr (iii) , (Q) rarr (iv) (R ) rarr (i) , (S ) rarr (ii)`
B. `(P) rarr (iv) , ( Q) rarr (i) , ( R) rarr (ii) , ( S) rarr (iii)`
C. `(P ) rarr (iii) , (Q) rarr (i) , (R ) rarr (iv) , (S ) rarr (ii)`
D. `(P ) rarr (i) , (Q) rarr (iii) , ( R) rarr (iv) , (S ) rarr (ii)`

Answer

To solve the problem of matching the rational forms of the given repeating decimals, we will convert each decimal into its fractional form step by step.

### Step 1: Convert \(0.\overline{32}\) to a Fraction
Let \(x = 0.\overline{32}\).

1. Since the repeating part has 2 digits, multiply both sides by 100:
\[ 100x = 32.\overline{32} \]
2. Now, subtract the original equation from this new equation:
\[ 100x - x = 32.\overline{32} - 0.\overline{32} \]
This simplifies to:
\[ 99x = 32 \]
3. Solve for \(x\):
\[ x = \frac{32}{99} \]

### Step 2: Convert \(0.2\overline{54}\) to a Fraction
Let \(y = 0.2\overline{54}\).

1. Rewrite \(y\) as \(y = 0.2545454...\).
2. Multiply both sides by 1000 (to shift the decimal point three places to the right):
\[ 1000y = 254.5454... \]
3. Multiply both sides by 10 (to shift the decimal point one more place):
\[ 10y = 2.545454... \]
4. Now, subtract the second equation from the first:
\[ 1000y - 10y = 254.5454... - 2.545454... \]
This simplifies to:
\[ 990y = 252 \]
5. Solve for \(y\):
\[ y = \frac{252}{990} \]
Simplifying gives:
\[ y = \frac{14}{55} \]

### Step 3: Convert \(0.\overline{12}\) to a Fraction
Let \(z = 0.\overline{12}\).

1. Since the repeating part has 2 digits, multiply both sides by 100:
\[ 100z = 12.\overline{12} \]
2. Now, subtract the original equation from this new equation:
\[ 100z - z = 12.\overline{12} - 0.\overline{12} \]
This simplifies to:
\[ 99z = 12 \]
3. Solve for \(z\):
\[ z = \frac{12}{99} = \frac{4}{33} \]

### Step 4: Convert \(0.2\overline{4}\) to a Fraction
Let \(w = 0.2\overline{4}\).

1. Rewrite \(w\) as \(w = 0.244444...\).
2. Multiply both sides by 10:
\[ 10w = 2.44444... \]
3. Multiply both sides by 100:
\[ 100w = 24.4444... \]
4. Now, subtract the first equation from the second:
\[ 100w - 10w = 24.4444... - 2.4444... \]
This simplifies to:
\[ 90w = 22 \]
5. Solve for \(w\):
\[ w = \frac{22}{90} = \frac{11}{45} \]

### Summary of Results
- \(0.\overline{32} = \frac{32}{99}\) (matches with (iii))
- \(0.2\overline{54} = \frac{14}{55}\) (matches with (i))
- \(0.\overline{12} = \frac{11}{90}\) (matches with (iv))
- \(0.2\overline{4} = \frac{11}{45}\) (matches with (ii))

### Final Matching
- (P) \(0.\overline{32}\) matches with (iii) \(\frac{32}{99}\)
- (Q) \(0.2\overline{54}\) matches with (i) \(\frac{14}{55}\)
- (R) \(0.\overline{12}\) matches with (iv) \(\frac{11}{90}\)
- (S) \(0.2\overline{4}\) matches with (ii) \(\frac{11}{45}\)