Which is smaller `(-5)/(6)` and `(-7)/(12)`?

Options

A. `(-5)/(6)`
B. `(-7)/(12)`
C. Cannot be compared
D. both are equal

Answer

To determine which of the two rational numbers, \(-\frac{5}{6}\) and \(-\frac{7}{12}\), is smaller, we can follow these steps:

### Step 1: Identify the two fractions
We have two fractions to compare:
1. \(-\frac{5}{6}\)
2. \(-\frac{7}{12}\)

### Step 2: Find a common denominator
The denominators of the two fractions are 6 and 12. To compare them easily, we need a common denominator. The least common multiple (LCM) of 6 and 12 is 12.

### Step 3: Convert \(-\frac{5}{6}\) to have the common denominator
To convert \(-\frac{5}{6}\) to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 2:
\[ -\frac{5}{6} = -\frac{5 \times 2}{6 \times 2} = -\frac{10}{12} \]

### Step 4: Write both fractions with the common denominator
Now we have:
1. \(-\frac{10}{12}\) (which is the converted form of \(-\frac{5}{6}\))
2. \(-\frac{7}{12}\)

### Step 5: Compare the numerators
Now that both fractions have the same denominator, we can compare the numerators:
- The numerators are \(-10\) and \(-7\).

### Step 6: Determine which numerator is smaller
On the number line, \(-10\) is to the left of \(-7\), which means \(-10\) is smaller than \(-7\). Therefore:
\[ -\frac{10}{12} < -\frac{7}{12} \]

### Conclusion
Thus, \(-\frac{5}{6}\) is smaller than \(-\frac{7}{12}\).

**Final Answer:**
\(-\frac{5}{6}\) is smaller than \(-\frac{7}{12}\).

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