Problem 31 - Elementary and Intermediate Algebra

Question

Multiply. $$ \frac{2}{3} \cdot\left(-\frac{3}{5}\right) $$

Step-by-Step Solution

Answer

\(\frac{-2}{5}\)

Step 1: Identify the numerators and denominators

Write down the numerators and denominators of both fractions: The first fraction is \(\frac{2}{3}\), where 2 is the numerator and 3 is the denominator. The second fraction is \( -\frac{3}{5} \), where -3 is the numerator and 5 is the denominator.

Step 2: Multiply the numerators

Multiply the numerators of both fractions:\[ 2 \times (-3) = -6 \]

Step 3: Multiply the denominators

Multiply the denominators of both fractions:\[ 3 \times 5 = 15 \]

Step 4: Simplify the fraction (if possible)

Combine the results from steps 2 and 3 into one fraction:\[\frac{-6}{15}\]Check if this fraction can be simplified. Both numerator and denominator can be divided by 3:\[\frac{-6 \div 3}{15 \div 3} = \frac{-2}{5}\]

Key Concepts

Fractions

Fractions are a way to represent a part of a whole. They consist of two main parts: a numerator and a denominator.
The numerator represents how many parts are being considered, while the denominator represents the total number of equal parts the whole is divided into.

For example, in the fraction \(\frac{2}{3}\), 2 is the numerator, and 3 is the denominator.
Fractions can be used in various operations such as addition, subtraction, multiplication, and division.
Understanding how fractions work is essential for solving mathematical problems that involve parts of a whole.

Numerators and Denominators

In a fraction, the numerator is the top number, and the denominator is the bottom number.
For instance, in \(\frac{2}{3}\), 2 is the numerator, and 3 is the denominator.
When we multiply fractions, we multiply the numerators together and the denominators together.
This helps in combining the parts correctly.
As shown in the exercise, to multiply \(\frac{2}{3} \times -\frac{3}{5}\), we first focus on the numerators 2 and -3. Multiplying them together gives us -6.
Next, we focus on the denominators 3 and 5. Multiplying these gives us 15.
The resulting fraction before simplification is \- \frac{6}{15}\.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible.
This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
By simplifying, we can make fractions easier to understand and work with.
Let's take the fraction \(\frac{-6}{15}\) from our previous result. We look for the GCD of 6 and 15. Here, the GCD is 3.
We divide both the numerator and the denominator by 3:

• \( -6 \div 3 = -2 \)
• \( 15 \div 3 = 5 \)

This gives us the simplified fraction: \(\frac{-2}{5}\).
By simplifying fractions, we make calculations more manageable and results more interpretable.