If a, b and c are real numbers, such that` a lt b and c lt 0,` then of the statements which is true?

Options

A. `(a)/(c) lt (b)/(c)`
B. `ac lt bc`
C. `(c)/(a) gt (c)/(b)`
D. `ac gt bc`

Answer

To solve the problem, we start with the given conditions and analyze the implications of multiplying or dividing by a negative number.

### Step-by-Step Solution:

1. **Understand the Given Conditions:**
We are given that \( a < b \) and \( c < 0 \). This means:
- \( a \) is less than \( b \).
- \( c \) is a negative number.

2. **Multiplying by a Negative Number:**
When we multiply both sides of the inequality \( a < b \) by \( c \) (which is negative), the direction of the inequality will change. Therefore:
\[ a \cdot c > b \cdot c \]
This means that the product of \( a \) and \( c \) is greater than the product of \( b \) and \( c \).

3. **Dividing by a Negative Number:**
Similarly, if we divide both sides of the inequality \( a < b \) by \( c \) (again, since \( c \) is negative), the direction of the inequality will also change:
\[ \frac{a}{c} > \frac{b}{c} \]
This means that the division of \( a \) by \( c \) is greater than the division of \( b \) by \( c \).

4. **Conclusion:**
Both operations (multiplication and division by \( c \)) lead us to the conclusion that:
- \( a \cdot c > b \cdot c \)
- \( \frac{a}{c} > \frac{b}{c} \)

5. **Identifying the True Statement:**
From the above analysis, we can conclude that the true statements among the options provided would be:
- \( a \cdot c > b \cdot c \)
- \( \frac{a}{c} > \frac{b}{c} \)