Divide the sum of `(-12)/(5) and (-18)/(15)` by their difference.

Options

A. 3
B. `-9`
C. `-7`
D. 5

Answer

To solve the problem of dividing the sum of \(-\frac{12}{5}\) and \(-\frac{18}{15}\) by their difference, we can follow these steps:

### Step 1: Find the sum of the two rational numbers.
We need to calculate:
\[ -\frac{12}{5} + -\frac{18}{15} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 5 and 15 is 15. We can convert \(-\frac{12}{5}\) to have a denominator of 15:
\[ -\frac{12}{5} = -\frac{12 \times 3}{5 \times 3} = -\frac{36}{15} \]

Now we can add:
\[ -\frac{36}{15} + -\frac{18}{15} = -\frac{36 + 18}{15} = -\frac{54}{15} \]

### Step 2: Find the difference of the two rational numbers.
Next, we calculate:
\[ -\frac{12}{5} - -\frac{18}{15} \]
This simplifies to:
\[ -\frac{12}{5} + \frac{18}{15} \]

Again, we convert \(-\frac{12}{5}\) to have a denominator of 15:
\[ -\frac{12}{5} = -\frac{36}{15} \]

Now we can find the difference:
\[ -\frac{36}{15} + \frac{18}{15} = -\frac{36 - 18}{15} = -\frac{18}{15} \]

### Step 3: Divide the sum by the difference.
Now we need to divide the sum by the difference:
\[ \frac{-\frac{54}{15}}{-\frac{18}{15}} \]

When dividing fractions, we multiply by the reciprocal of the denominator:
\[ = -\frac{54}{15} \times -\frac{15}{18} \]

The negatives cancel out:
\[ = \frac{54 \times 15}{15 \times 18} \]

The \(15\) in the numerator and denominator cancels out:
\[ = \frac{54}{18} \]

Now we simplify:
\[ = 3 \]

### Final Answer:
The value of the expression is \(3\).

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