Express each of the following recurring decimals into the rational number :
`(i)0.bar(5)" "(ii)2.bar(4)" "(iii)1.bar(12)" "(iv)2.7bar(39)" "(v)0.bar(516)" "(vi)3.7bar(148)`

Explanation

To express each of the given recurring decimals into rational numbers, we will follow a systematic approach for each part.

### (i) Convert \(0.\overline{5}\)

1. Let \(x = 0.\overline{5}\).
2. This means \(x = 0.5555...\).
3. Multiply both sides by 10:
\[ 10x = 5.5555... \]
4. Now, subtract the first equation from the second:
\[ 10x - x = 5.5555... - 0.5555... \]
This simplifies to:
\[ 9x = 5 \]
5. Therefore,
\[ x = \frac{5}{9} \]

### (ii) Convert \(2.\overline{4}\)

1. Let \(x = 2.\overline{4}\).
2. This means \(x = 2.4444...\).
3. Multiply both sides by 10:
\[ 10x = 24.4444... \]
4. Subtract the first equation from the second:
\[ 10x - x = 24.4444... - 2.4444... \]
This simplifies to:
\[ 9x = 22 \]
5. Therefore,
\[ x = \frac{22}{9} \]

### (iii) Convert \(1.\overline{12}\)

1. Let \(x = 1.\overline{12}\).
2. This means \(x = 1.121212...\).
3. Multiply both sides by 100:
\[ 100x = 112.1212... \]
4. Subtract the first equation from the second:
\[ 100x - x = 112.1212... - 1.1212... \]
This simplifies to:
\[ 99x = 111 \]
5. Therefore,
\[ x = \frac{111}{99} = \frac{37}{33} \quad (\text{after simplification}) \]

### (iv) Convert \(2.7\overline{39}\)

1. Let \(x = 2.7\overline{39}\).
2. This means \(x = 2.7393939...\).
3. Multiply both sides by 10:
\[ 10x = 27.393939... \]
4. Multiply both sides by 1000:
\[ 1000x = 2739.393939... \]
5. Subtract the second equation from the first:
\[ 1000x - 10x = 2739.393939... - 27.393939... \]
This simplifies to:
\[ 990x = 2712 \]
6. Therefore,
\[ x = \frac{2712}{990} = \frac{452}{165} \quad (\text{after simplification}) \]

### (v) Convert \(0.\overline{516}\)

1. Let \(x = 0.\overline{516}\).
2. This means \(x = 0.516516...\).
3. Multiply both sides by 1000:
\[ 1000x = 516.516516... \]
4. Subtract the first equation from the second:
\[ 1000x - x = 516.516516... - 0.516516... \]
This simplifies to:
\[ 999x = 516 \]
5. Therefore,
\[ x = \frac{516}{999} = \frac{172}{333} \quad (\text{after simplification}) \]

### (vi) Convert \(3.7\overline{148}\)

1. Let \(x = 3.7\overline{148}\).
2. This means \(x = 3.714814814...\).
3. Multiply both sides by 10:
\[ 10x = 37.14814814... \]
4. Multiply both sides by 10000:
\[ 10000x = 37148.14814814... \]
5. Subtract the second equation from the first:
\[ 10000x - 10x = 37148.14814814... - 37.14814814... \]
This simplifies to:
\[ 9990x = 37111 \]
6. Therefore,
\[ x = \frac{37111}{9990} = \frac{1003}{270} \quad (\text{after simplification}) \]

### Summary of Results:
1. \(0.\overline{5} = \frac{5}{9}\)
2. \(2.\overline{4} = \frac{22}{9}\)
3. \(1.\overline{12} = \frac{37}{33}\)
4. \(2.7\overline{39} = \frac{452}{165}\)
5. \(0.\overline{516} = \frac{172}{333}\)
6. \(3.7\overline{148} = \frac{1003}{270}\)