Find four rational numbers equivalent to each of the following:
(i) `3/10`, (ii) `-5/9`, (iii) `6/(-13)`, (iv) 9, (v) `-1`

Answer

To find four rational numbers equivalent to each of the given numbers, we will multiply each number by different fractions that equal 1 (like 2/2, 3/3, etc.). Let's solve each part step by step.

### (i) For `3/10`:
1. Multiply `3/10` by `2/2`:
\[ \frac{3}{10} \times \frac{2}{2} = \frac{6}{20} \]

2. Multiply `3/10` by `-2/2`:
\[ \frac{3}{10} \times \frac{-2}{2} = \frac{-6}{20} \]

3. Multiply `3/10` by `3/3`:
\[ \frac{3}{10} \times \frac{3}{3} = \frac{9}{30} \]

4. Multiply `3/10` by `4/4`:
\[ \frac{3}{10} \times \frac{4}{4} = \frac{12}{40} \]

**Equivalent rational numbers for `3/10` are:**
- `6/20`
- `-6/20`
- `9/30`
- `12/40`

### (ii) For `-5/9`:
1. Multiply `-5/9` by `2/2`:
\[ \frac{-5}{9} \times \frac{2}{2} = \frac{-10}{18} \]

2. Multiply `-5/9` by `3/3`:
\[ \frac{-5}{9} \times \frac{3}{3} = \frac{-15}{27} \]

3. Multiply `-5/9` by `4/4`:
\[ \frac{-5}{9} \times \frac{4}{4} = \frac{-20}{36} \]

4. Multiply `-5/9` by `5/5`:
\[ \frac{-5}{9} \times \frac{5}{5} = \frac{-25}{45} \]

**Equivalent rational numbers for `-5/9` are:**
- `-10/18`
- `-15/27`
- `-20/36`
- `-25/45`

### (iii) For `6/(-13)`:
1. Multiply `6/(-13)` by `2/2`:
\[ \frac{6}{-13} \times \frac{2}{2} = \frac{12}{-26} \]

2. Multiply `6/(-13)` by `3/3`:
\[ \frac{6}{-13} \times \frac{3}{3} = \frac{18}{-39} \]

3. Multiply `6/(-13)` by `4/4`:
\[ \frac{6}{-13} \times \frac{4}{4} = \frac{24}{-52} \]

4. Multiply `6/(-13)` by `5/5`:
\[ \frac{6}{-13} \times \frac{5}{5} = \frac{30}{-65} \]

**Equivalent rational numbers for `6/(-13)` are:**
- `12/(-26)`
- `18/(-39)`
- `24/(-52)`
- `30/(-65)`

### (iv) For `9`:
1. Write `9` as `9/1` and multiply by `2/2`:
\[ \frac{9}{1} \times \frac{2}{2} = \frac{18}{2} \]

2. Multiply `9/1` by `3/3`:
\[ \frac{9}{1} \times \frac{3}{3} = \frac{27}{3} \]

3. Multiply `9/1` by `4/4`:
\[ \frac{9}{1} \times \frac{4}{4} = \frac{36}{4} \]

4. Multiply `9/1` by `5/5`:
\[ \frac{9}{1} \times \frac{5}{5} = \frac{45}{5} \]

**Equivalent rational numbers for `9` are:**
- `18/2`
- `27/3`
- `36/4`
- `45/5`

### (v) For `-1`:
1. Write `-1` as `-1/1` and multiply by `2/2`:
\[ \frac{-1}{1} \times \frac{2}{2} = \frac{-2}{2} \]

2. Multiply `-1/1` by `3/3`:
\[ \frac{-1}{1} \times \frac{3}{3} = \frac{-3}{3} \]

3. Multiply `-1/1` by `4/4`:
\[ \frac{-1}{1} \times \frac{4}{4} = \frac{-4}{4} \]

4. Multiply `-1/1` by `5/5`:
\[ \frac{-1}{1} \times \frac{5}{5} = \frac{-5}{5} \]

**Equivalent rational numbers for `-1` are:**
- `-2/2`
- `-3/3`
- `-4/4`
- `-5/5`

### Summary of Equivalent Rational Numbers:
- For `3/10`: `6/20`, `-6/20`, `9/30`, `12/40`
- For `-5/9`: `-10/18`, `-15/27`, `-20/36`, `-25/45`
- For `6/(-13)`: `12/(-26)`, `18/(-39)`, `24/(-52)`, `30/(-65)`
- For `9`: `18/2`, `27/3`, `36/4`, `45/5`
- For `-1`: `-2/2`, `-3/3`, `-4/4`, `-5/5`