Problem 36
Question
For exercises \(23-74\), evaluate. $$ \left(-\frac{2}{3}\right)\left(-\frac{1}{2}\right) $$
Step-by-Step Solution
Verified Answer
\(\frac{1}{3}\)
1Step 1: Identify the Multiplicands
Here the two fractions to be multiplied are \(-\frac{2}{3}\) and \(-\frac{1}{2}\).
2Step 2: Multiply the Numerators
Multiply the numerators of the fractions: \(-2 \times -1 = 2\).
3Step 3: Multiply the Denominators
Multiply the denominators of the fractions: \(3 \times 2 = 6\).
4Step 4: Combine the Results
The result of the multiplication is \(\frac{2}{6}\).
5Step 5: Simplify the Fraction
Simplify the fraction \(\frac{2}{6}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2: \(\frac{2 \div 2}{6 \div 2} = \frac{1}{3}\).
Key Concepts
Negative NumbersNumerators and DenominatorsFraction Simplification
Negative Numbers
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: \[ \underline{\phantom{xxx}} -\frac{2}{3} \times -\frac{1}{2} \underline{\phantom{xxx}} \]
Both fractions have negative signs, which cancel each other out, resulting in a positive fraction after the multiplication of the numerators and denominators.
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: \[ \underline{\phantom{xxx}} -\frac{2}{3} \times -\frac{1}{2} \underline{\phantom{xxx}} \]
Both fractions have negative signs, which cancel each other out, resulting in a positive fraction after the multiplication of the numerators and denominators.
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.
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.
Fraction 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: \[ \frac{2 \underline{\phantom{xxx}} \text{\textdiv} \underline{\phantom{xxx}} 2}{6 \underline{\phantom{xxx}} \text{\textdiv} \underline{\phantom{xxx}} 2} = \frac{1}{3} \]
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.
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: \[ \frac{2 \underline{\phantom{xxx}} \text{\textdiv} \underline{\phantom{xxx}} 2}{6 \underline{\phantom{xxx}} \text{\textdiv} \underline{\phantom{xxx}} 2} = \frac{1}{3} \]
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.
Other exercises in this chapter
Problem 35
For exercises 1-80, evaluate. $$ 6+5 \cdot 0 $$
View solution Problem 36
$$ \text { Round } 0.0053 \text { to the nearest thousandth. } $$
View solution Problem 36
For exercises 1-80, evaluate. $$ 9+2 \cdot 0 $$
View solution Problem 37
$$ \text { Round } 8.01 \text { to the nearest tenth. } $$
View solution