Convert the following fractions into equivalent like fractions :
(i) `7/12, 5/21 " (ii) " 4/5, 7/15, 31/20`.

Answer

To convert the given fractions into equivalent like fractions, we will follow these steps for each part of the question.

### Part (i): Convert `7/12` and `5/21` into equivalent like fractions.

**Step 1: Find the LCM of the denominators.**
- The denominators are 12 and 21.
- To find the LCM, we can list the multiples or use the prime factorization method.
- Prime factorization:
- 12 = 2² * 3
- 21 = 3 * 7
- The LCM is found by taking the highest power of each prime:
- LCM = 2² * 3 * 7 = 4 * 3 * 7 = 84.

**Step 2: Convert each fraction to have the LCM as the denominator.**
- For `7/12`, we need to convert it to a fraction with a denominator of 84:
- Multiply both the numerator and denominator by 7 (since 12 * 7 = 84).
- \( \frac{7}{12} \times \frac{7}{7} = \frac{49}{84} \).

- For `5/21`, we need to convert it to a fraction with a denominator of 84:
- Multiply both the numerator and denominator by 4 (since 21 * 4 = 84).
- \( \frac{5}{21} \times \frac{4}{4} = \frac{20}{84} \).

**Step 3: Write the equivalent fractions.**
- The equivalent like fractions are:
- \( \frac{49}{84} \) and \( \frac{20}{84} \).

### Part (ii): Convert `4/5`, `7/15`, and `31/20` into equivalent like fractions.

**Step 1: Find the LCM of the denominators.**
- The denominators are 5, 15, and 20.
- Prime factorization:
- 5 = 5
- 15 = 3 * 5
- 20 = 2² * 5
- The LCM is found by taking the highest power of each prime:
- LCM = 2² * 3 * 5 = 4 * 3 * 5 = 60.

**Step 2: Convert each fraction to have the LCM as the denominator.**
- For `4/5`, we need to convert it to a fraction with a denominator of 60:
- Multiply both the numerator and denominator by 12 (since 5 * 12 = 60).
- \( \frac{4}{5} \times \frac{12}{12} = \frac{48}{60} \).

- For `7/15`, we need to convert it to a fraction with a denominator of 60:
- Multiply both the numerator and denominator by 4 (since 15 * 4 = 60).
- \( \frac{7}{15} \times \frac{4}{4} = \frac{28}{60} \).

- For `31/20`, we need to convert it to a fraction with a denominator of 60:
- Multiply both the numerator and denominator by 3 (since 20 * 3 = 60).
- \( \frac{31}{20} \times \frac{3}{3} = \frac{93}{60} \).

**Step 3: Write the equivalent fractions.**
- The equivalent like fractions are:
- \( \frac{48}{60} \), \( \frac{28}{60} \), and \( \frac{93}{60} \).

### Final Answers:
- (i) \( \frac{49}{84}, \frac{20}{84} \)
- (ii) \( \frac{48}{60}, \frac{28}{60}, \frac{93}{60} \)