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When dealing with negative numbers, especially in fraction multiplication, it's important to understand how the signs affect the result. A negative sign in front of a fraction indicates that the entire value of the fraction is negative.
When you multiply two negative numbers, the result is positive. This is because multiplying two negatives cancels out the negativity.
For example:
\[ (-2) \times (-3) = 6 \]
In the given exercise, you multiply two negative fractions: \[ \text{ } -\frac{2}{3} \times -\frac{1}{2} \text{ } \]
Both fractions have negative signs, which cancel each other out, resulting in a positive fraction after the multiplication of the numerators and denominators.

In fractions, the numerator is the top number, and the denominator is the bottom number.
The numerator represents how many parts you have, while the denominator shows how many parts make up a whole.
When multiplying fractions, you multiply the numerators together and the denominators together.
For the given example: \[ -\frac{2}{3} \times -\frac{1}{2} \]
Multiply the numerators: \[ -2 \times -1 = 2 \]
Multiply the denominators: \[ 3 \times 2 = 6 \]
This gives us the fraction: \[ \frac{2}{6} \]
When the numerators and denominators are multiplied separately, the problem becomes much simpler, and you can move on to simplification.

Simplifying a fraction means reducing it to its lowest terms. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).
The GCD of two numbers is the highest number that divides both of them without leaving a remainder.
For the fraction \[ \frac{2}{6} \], the GCD of 2 and 6 is 2.
To simplify:
Divide both the numerator and the denominator by their GCD: \[ \frac{2 \text{ } \text{\textdiv} \text{ } 2}{6 \text{ } \text{\textdiv} \text{ } 2} = \frac{1}{3} \]
This results in the simplified fraction \[ \frac{1}{3} \]
Simplifying fractions makes them easier to understand and use in further calculations. It helps in seeing the most reduced form of the fraction and aids clarity.