Which of the two rational number `(2)/(-3)` and `(-4)/(5)` is greater ?

Options

A. `2/(-3)`
B. `(-4)/5`
C. `"both are equal"`
D. `"can not be compared"`

Answer

To determine which of the two rational numbers \( \frac{2}{-3} \) and \( \frac{-4}{5} \) is greater, we can follow these steps:

### Step 1: Identify the Rational Numbers
We have two rational numbers:
1. \( \frac{2}{-3} \)
2. \( \frac{-4}{5} \)

### Step 2: Convert to a Common Denominator
To compare these two fractions, we need to have a common denominator. The denominators are -3 and 5. The least common multiple (LCM) of 3 and 5 is 15.

### Step 3: Rewrite Each Fraction with the Common Denominator
Now we will convert each fraction to have a denominator of 15.

1. For \( \frac{2}{-3} \):
- Multiply both the numerator and the denominator by 5:
\[ \frac{2 \times 5}{-3 \times 5} = \frac{10}{-15} = \frac{-10}{15} \]

2. For \( \frac{-4}{5} \):
- Multiply both the numerator and the denominator by 3:
\[ \frac{-4 \times 3}{5 \times 3} = \frac{-12}{15} \]

### Step 4: Compare the Numerators
Now we have:
- \( \frac{-10}{15} \)
- \( \frac{-12}{15} \)

Since both fractions have the same denominator, we can compare the numerators directly:
- The numerator of \( \frac{-10}{15} \) is -10.
- The numerator of \( \frac{-12}{15} \) is -12.

### Step 5: Determine Which is Greater
On the number line, -10 is to the right of -12, which means:
\[ -10 > -12 \]
Thus, \( \frac{-10}{15} \) (which is \( \frac{2}{-3} \)) is greater than \( \frac{-12}{15} \) (which is \( \frac{-4}{5} \)).

### Final Answer
Therefore, \( \frac{2}{-3} \) is greater than \( \frac{-4}{5} \).

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