Arrange the numbers `(-3)/(5), (7)/(-10) and (-5)/(8)` in ascending order.

Options

A. `(-3)/5 lt 7/(-10) lt (-5)/8`
B. `(7)/(-10) lt (-5)/(8) lt (-3)/(5)`
C. `(-5)/8 lt 7/(-10) lt (-3)/5`
D. none of these

Answer

To arrange the numbers \(-\frac{3}{5}\), \(\frac{7}{-10}\), and \(-\frac{5}{8}\) in ascending order, we will follow these steps:

### Step 1: Identify the numbers
The numbers we need to arrange are:
1. \(-\frac{3}{5}\)
2. \(\frac{7}{-10} = -\frac{7}{10}\)
3. \(-\frac{5}{8}\)

### Step 2: Find a common denominator
The denominators of the fractions are 5, 10, and 8. We need to find the Least Common Multiple (LCM) of these denominators to convert all fractions to have the same denominator.

- The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, ...
- The multiples of 10 are: 10, 20, 30, 40, ...
- The multiples of 8 are: 8, 16, 24, 32, 40, ...

The LCM of 5, 10, and 8 is 40.

### Step 3: Convert each fraction to have the common denominator
Now, we will convert each fraction to have a denominator of 40.

1. For \(-\frac{3}{5}\):
\[ -\frac{3}{5} = -\frac{3 \times 8}{5 \times 8} = -\frac{24}{40} \]

2. For \(-\frac{7}{10}\):
\[ -\frac{7}{10} = -\frac{7 \times 4}{10 \times 4} = -\frac{28}{40} \]

3. For \(-\frac{5}{8}\):
\[ -\frac{5}{8} = -\frac{5 \times 5}{8 \times 5} = -\frac{25}{40} \]

### Step 4: Compare the fractions
Now we have:
- \(-\frac{24}{40}\)
- \(-\frac{28}{40}\)
- \(-\frac{25}{40}\)

To arrange them in ascending order, we compare the numerators:
- \(-28 < -25 < -24\)

### Step 5: Write the fractions in ascending order
Thus, in ascending order, we have:
\[ -\frac{28}{40} < -\frac{25}{40} < -\frac{24}{40} \]

Converting back to the original fractions:
\[ -\frac{7}{10} < -\frac{5}{8} < -\frac{3}{5} \]

### Final Answer:
The numbers in ascending order are:
\[ -\frac{7}{10}, -\frac{5}{8}, -\frac{3}{5} \]

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