Write each of the following rational numbers in standard form :
(i) `56/32`, (ii) `16/(-40)`, (iii) `-36/(54)`, (iv) `-22/-77`
(v) `78/(-65)`, (vi) `-5/114`, (vi) `-69/115`, (vii) `155/(-217)`

Answer

To write each of the given rational numbers in standard form, we need to simplify them to their lowest terms. The standard form of a rational number is when the numerator and denominator have no common factors other than 1.

Let's solve each part step by step:

### (i) \( \frac{56}{32} \)

1. **Find the GCD (Greatest Common Divisor)** of 56 and 32.
- The factors of 56 are: 1, 2, 4, 7, 8, 14, 28, 56
- The factors of 32 are: 1, 2, 4, 8, 16, 32
- The GCD is 8.

2. **Divide both the numerator and denominator by the GCD**:
\[ \frac{56 \div 8}{32 \div 8} = \frac{7}{4} \]

### (ii) \( \frac{16}{-40} \)

1. **Find the GCD of 16 and 40**.
- The GCD is 8.

2. **Divide both the numerator and denominator by the GCD**:
\[ \frac{16 \div 8}{-40 \div 8} = \frac{2}{-5} = -\frac{2}{5} \]

### (iii) \( \frac{-36}{54} \)

1. **Find the GCD of 36 and 54**.
- The GCD is 18.

2. **Divide both the numerator and denominator by the GCD**:
\[ \frac{-36 \div 18}{54 \div 18} = \frac{-2}{3} \]

### (iv) \( \frac{-22}{-77} \)

1. **The negatives cancel out**:
\[ \frac{22}{77} \]

2. **Find the GCD of 22 and 77**.
- The GCD is 11.

3. **Divide both the numerator and denominator by the GCD**:
\[ \frac{22 \div 11}{77 \div 11} = \frac{2}{7} \]

### (v) \( \frac{78}{-65} \)

1. **Find the GCD of 78 and 65**.
- The GCD is 13.

2. **Divide both the numerator and denominator by the GCD**:
\[ \frac{78 \div 13}{-65 \div 13} = \frac{6}{-5} = -\frac{6}{5} \]

### (vi) \( \frac{-5}{114} \)

1. **Check for common factors**. The GCD is 1.

2. **No simplification is possible**:
\[ \frac{-5}{114} \]

### (vii) \( \frac{-69}{115} \)

1. **Find the GCD of 69 and 115**.
- The GCD is 23.

2. **Divide both the numerator and denominator by the GCD**:
\[ \frac{-69 \div 23}{115 \div 23} = \frac{-3}{5} \]

### (viii) \( \frac{155}{-217} \)

1. **Find the GCD of 155 and 217**.
- The GCD is 31.

2. **Divide both the numerator and denominator by the GCD**:
\[ \frac{155 \div 31}{-217 \div 31} = \frac{5}{-7} = -\frac{5}{7} \]

### Summary of Standard Forms:
1. \( \frac{56}{32} = \frac{7}{4} \)
2. \( \frac{16}{-40} = -\frac{2}{5} \)
3. \( \frac{-36}{54} = -\frac{2}{3} \)
4. \( \frac{-22}{-77} = \frac{2}{7} \)
5. \( \frac{78}{-65} = -\frac{6}{5} \)
6. \( \frac{-5}{114} = \frac{-5}{114} \)
7. \( \frac{-69}{115} = -\frac{3}{5} \)
8. \( \frac{155}{-217} = -\frac{5}{7} \)