Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$\left(-\frac{4}{5}\right)(-30)$$

Step-by-Step Solution

Answer

Therefore, the product of \(\left(-\frac{4}{5}\right)(-30)\) is 24.

Step 1: Handling the Negative Signs

Initially, focus on the negative signs. In multiplication, the product of two negative numbers gives a positive number. So, the negative signs of both numbers cancel each other out and the product of these two numbers then becomes positive.

Step 2: Multiplication with the Whole Number

Next, perform the multiplication of the fraction \(-4/5\) with the whole number \(30\). To carry out this fraction multiplication, convert the whole number into a fraction by simply putting it over \(1\), making it \(30/1\). So, the multiplication of \(-4/5\) and \(30/1\) gives \(-120/5\).

Step 3: Simplify the Result

Then, simplify the obtained result \(-120/5\) to obtain the final answer. Dividing -120 by 5 gives -24.

Key Concepts

Negative Numbers

Understanding how negative numbers work is crucial in algebra. When we talk about negative numbers, we're referring to numbers less than zero. These numbers are important not only in math but also in real-life contexts, like temperatures below freezing or debts.

A key rule with negative numbers is their interaction in multiplication. When you multiply two negative numbers, the result is positive. For example, \(-1 imes -1 = 1\). This is because two negatives cancel each other out. In our exercise, \(-\frac{4}{5}\) and \(-30\) are both negative, so their product is positive.

Fraction Multiplication

Fraction multiplication might appear tricky at first, but it follows a straightforward process. The main steps are multiplying the numerators and then the denominators.

To illustrate, if you have two fractions, say \(\frac{a}{b}\) and \(\frac{c}{d}\), the multiplication is done as follows: \(\frac{a \times c}{b \times d}\). This process keeps the fractions intact and simplifies problem-solving.

• Step 1: Multiply the numerators.
• Step 2: Multiply the denominators.
• Step 3: Simplify if possible.

Simplification

Simplification is the process of reducing an expression to its simplest form, making it easier to understand or solve. This involves reducing fractions by finding the greatest common divisor (GCD) of the numerator and denominator.

In our exercise, after multiplying, we got the fraction \(-\frac{120}{5}\). The GCD of 120 and 5 is 5. Therefore, dividing the numerator and the denominator by this number simplifies the expression to \(-24\). Simplifying fractions helps in making calculations neat and manageable.

Multiplying Fractions

When you multiply fractions, it's important to break down each step logically. For this, fractions interact like regular numerical multiplication but require additional attention to detail. To multiply fractions accurately:

• First, convert any whole numbers into fractions by placing them over 1.
• Next, multiply the numerators together.
• Then, multiply the denominators.
• Finally, simplify the resulting fraction if possible.

Let's apply this to our exercise: \(\left(-\frac{4}{5}\right) \times (30/1)\) gives us a fraction — \(-\frac{120}{5}\) — which we then simplify to \(-24\). Understanding these steps ensures accuracy in combining fractions with whole numbers.