Problem 47 - Algebra for College Students

Question

Perform the following operations with real numbers. $$\left(-\frac{3}{4}\right)\left(\frac{4}{5}\right)$$

Step-by-Step Solution

Answer

The product of \(-\frac{3}{4}\) and \(\frac{4}{5}\) is \(-\frac{3}{5}\).

Step 1: Identify the Operation

We are asked to multiply two fractions, \(-\frac{3}{4}\) and \(\frac{4}{5}\). In this operation, we multiply the numerators together and the denominators together.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \(-3\) and \(4\). The calculation is \(-3 \times 4 = -12\).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \(4\) and \(5\). The calculation is \(4 \times 5 = 20\).

Step 4: Simplify the Fraction

Combine the results from Steps 2 and 3 to form a new fraction: \(\frac{-12}{20}\). To simplify, find the greatest common divisor (GCD) of 12 and 20, which is 4, and divide both the numerator and denominator by 4: \(\frac{-12 \div 4}{20 \div 4} = \frac{-3}{5}\).

Key Concepts

Multiplying Fractions

When multiplying fractions, you follow an easy process. The key is to understand that both the numerators and denominators must be handled separately. Let’s break this down step-by-step for clarity.

• First, look at the numerators. You take the top numbers of each fraction and multiply them together. For example, if you have \(-\frac{3}{4}\) and \(\frac{4}{5}\), the numerators are \(-3\) and \(4\). You multiply these: \(-3 \times 4 = -12\).


• Next, multiply the denominators, which are the bottom parts of the fractions. In our example, the denominators are \(4\) and \(5\). Multiply these two: \(4 \times 5 = 20\).


• Finally, place your results into a fraction: the product of the numerators over the product of the denominators, giving us \(\frac{-12}{20}\).

This process of multiplying the top and bottom numbers separately is straightforward. It only requires careful multiplication and then moving on to simplification if possible.

Simplifying Fractions

Once you've multiplied fractions, you often end up with a fraction that can be simplified. Simplification makes your answer more understandable and easier to work with. Let’s see how to do this.

• First, look for the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that divides both numbers without leaving a remainder. In our example \(\frac{-12}{20}\), the GCD is \(4\).


• Next, divide both parts of your fraction by the GCD. So, divide \(-12\) and \(20\) by \(4\). You end up with \(-3\) and \(5\) respectively.


• This gives you a simplified fraction: \(\frac{-3}{5}\). It’s a more compact and user-friendly form of your answer.

Simplifying fractions helps prevent handling unnecessarily large numbers and keeps your math tidy and efficient.

Numerators and Denominators

Understanding the parts of a fraction, namely the numerator and the denominator, is crucial for operations like multiplication and simplification. Let's break these down.

• The numerator is the top part of the fraction. It indicates how many parts of the whole are being considered. For instance, in \(-\frac{3}{4}\), the numerator is \(-3\).


• The denominator, on the other hand, is the bottom part. It shows into how many equal parts the whole is divided. In our example, the denominator is \(4\).


• When multiplying, the numerators interact with each other, and similarly, the denominators interact with each other, forming a new fraction, like \(\frac{-12}{20}\) from our previous operation.

Recognizing these parts helps guide how to approach problems, ensuring operations are applied correctly and simplification is performed accurately.