Question

To order the expressions from least to greatest, let's first convert them into improper fractions so we can easily compare them.

1. **-4 3/7** = -4 + (3/7) = -\(\frac{28}{7}\) - \(\frac{3}{7}\) = -\(\frac{31}{7}\)
2. **-4 6/7** = -4 + (6/7) = -\(\frac{28}{7}\) - \(\frac{6}{7}\) = -\(\frac{34}{7}\)
3. **-5 3/7** = -5 + (3/7) = -\(\frac{35}{7}\) - \(\frac{3}{7}\) = -\(\frac{38}{7}\)
4. **4 2/7** = 4 + (2/7) = \(\frac{28}{7}\) + \(\frac{2}{7}\) = \(\frac{30}{7}\)
5. **5 1/5** = 5 + (1/5) = \(\frac{25}{5}\) + \(\frac{1}{5}\) = \(\frac{26}{5}\) = \(\frac{78}{15}\) (this value is approximately 5.2)

Now, let's compare the values:

- **-5 3/7 (least)**: -\(\frac{38}{7}\) (approximately -5.43)
- **-4 6/7**: -\(\frac{34}{7}\) (approximately -4.86)
- **-4 3/7**: -\(\frac{31}{7}\) (approximately -4.43)
- **4 2/7**: \(\frac{30}{7}\) (approximately 4.29)
- **5 1/5 (greatest)**: \(\frac{26}{5}\) (approximately 5.2)

Now, let's arrange them from least to greatest:

1. **-5 3/7**
2. **-4 6/7**
3. **-4 3/7**
4. **4 2/7**
5. **5 1/5**

So the correct order from least to greatest is:

-5 3/7
-4 6/7
-4 3/7
4 2/7
5 1/5

the last one is supposed to be 5 1/7

Thank you for the correction. Let's redo the ordering with the updated expression \(5 \frac{1}{7}\).

1. **-4 \(\frac{3}{7}\)** = -4 + \(\frac{3}{7}\) = -\(\frac{28}{7}\) - \(\frac{3}{7}\) = -\(\frac{31}{7}\)

2. **-4 \(\frac{6}{7}\)** = -4 + \(\frac{6}{7}\) = -\(\frac{28}{7}\) - \(\frac{6}{7}\) = -\(\frac{34}{7}\)

3. **-5 \(\frac{3}{7}\)** = -5 + \(\frac{3}{7}\) = -\(\frac{35}{7}\) - \(\frac{3}{7}\) = -\(\frac{38}{7}\)

4. **4 \(\frac{2}{7}\)** = 4 + \(\frac{2}{7}\) = \(\frac{28}{7}\) + \(\frac{2}{7}\) = \(\frac{30}{7}\)

5. **5 \(\frac{1}{7}\)** = 5 + \(\frac{1}{7}\) = \(\frac{35}{7}\) + \(\frac{1}{7}\) = \(\frac{36}{7}\)

Now, let's compare the values:

- **-5 \(\frac{3}{7}\)** \(\approx -5.43\) (-\(\frac{38}{7}\))
- **-4 \(\frac{6}{7}\)** \(\approx -4.86\) (-\(\frac{34}{7}\))
- **-4 \(\frac{3}{7}\)** \(\approx -4.43\) (-\(\frac{31}{7}\))
- **4 \(\frac{2}{7}\)** \(\approx 4.29\) (\(\frac{30}{7}\))
- **5 \(\frac{1}{7}\)** \(\approx 5.14\) (\(\frac{36}{7}\))

Now we can arrange them from least to greatest:

1. **-5 \(\frac{3}{7}\)**
2. **-4 \(\frac{6}{7}\)**
3. **-4 \(\frac{3}{7}\)**
4. **4 \(\frac{2}{7}\)**
5. **5 \(\frac{1}{7}\)**

So the final order from least to greatest is:

-5 \(\frac{3}{7}\)
-4 \(\frac{6}{7}\)
-4 \(\frac{3}{7}\)
4 \(\frac{2}{7}\)
5 \(\frac{1}{7}\)