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 convert the given recurring decimals into rational numbers, we can follow a systematic approach. Let's go through each part step by step:

### (i) Convert \(0.\overline{5}\) to a rational number:

1. Let \(x = 0.\overline{5}\).
2. Multiply both sides by 10: \(10x = 5.\overline{5}\).
3. Subtract the first equation from the second:
\[ 10x - x = 5.\overline{5} - 0.\overline{5} \implies 9x = 5 \]
4. Solve for \(x\):
\[ x = \frac{5}{9} \]

### (ii) Convert \(2.\overline{4}\) to a rational number:

1. Let \(x = 2.\overline{4}\).
2. Multiply both sides by 10: \(10x = 24.\overline{4}\).
3. Subtract the first equation from the second:
\[ 10x - x = 24.\overline{4} - 2.\overline{4} \implies 9x = 22 \]
4. Solve for \(x\):
\[ x = \frac{22}{9} \]

### (iii) Convert \(1.\overline{12}\) to a rational number:

1. Let \(x = 1.\overline{12}\).
2. Multiply both sides by 100: \(100x = 112.\overline{12}\).
3. Subtract the first equation from the second:
\[ 100x - x = 112.\overline{12} - 1.\overline{12} \implies 99x = 111 \]
4. Solve for \(x\):
\[ x = \frac{111}{99} = \frac{37}{33} \quad (\text{after simplification}) \]

### (iv) Convert \(2.7\overline{39}\) to a rational number:

1. Let \(x = 2.7\overline{39}\).
2. Multiply both sides by 10: \(10x = 27.\overline{39}\).
3. Multiply both sides by 1000: \(1000x = 2739.\overline{39}\).
4. Subtract the first equation from the second:
\[ 1000x - 10x = 2739.\overline{39} - 27.\overline{39} \implies 990x = 2712 \]
5. Solve for \(x\):
\[ x = \frac{2712}{990} = \frac{2712 \div 18}{990 \div 18} = \frac{151}{55} \quad (\text{after simplification}) \]

### (v) Convert \(0.\overline{516}\) to a rational number:

1. Let \(x = 0.\overline{516}\).
2. Multiply both sides by 1000: \(1000x = 516.\overline{516}\).
3. Subtract the first equation from the second:
\[ 1000x - x = 516.\overline{516} - 0.\overline{516} \implies 999x = 516 \]
4. Solve for \(x\):
\[ x = \frac{516}{999} \]

### (vi) Convert \(3.7\overline{148}\) to a rational number:

1. Let \(x = 3.7\overline{148}\).
2. Multiply both sides by 10: \(10x = 37.\overline{148}\).
3. Multiply both sides by 1000: \(1000x = 37148.\overline{148}\).
4. Subtract the first equation from the second:
\[ 1000x - 10x = 37148.\overline{148} - 37.\overline{148} \implies 990x = 37111 \]
5. Solve for \(x\):
\[ x = \frac{37111}{990} \]

### 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. \(2.7\overline{39} = \frac{151}{55}\)
5. \(0.\overline{516} = \frac{516}{999}\)
6. \(3.7\overline{148} = \frac{37111}{990}\)