Problem 22 - Algebra for College Students

Question

Perform the following operations with real numbers. $$(-8)\left(\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

The result is \(\frac{-8}{3}\).

Step 1: Understand the Problem Statement

We are asked to perform a multiplication operation involving two numbers: \(-8\) and \(\frac{1}{3}\). This involves multiplying a negative integer with a positive fraction.

Step 2: Multiply the Numbers

To perform the multiplication, multiply the integer \(-8\) by the numerator of the fraction \(1\) and divide by the denominator \(3\). This gives us \(-8 \times \frac{1}{3} = \frac{-8 \times 1}{3} = \frac{-8}{3}\).

Step 3: Interpret the Result

The result of the multiplication is a fraction \(\frac{-8}{3}\). This is a negative fraction because multiplying a negative number by a positive number yields a negative result. The result can remain as an improper fraction or be converted to a mixed number.

Key Concepts

Multiplication of Fractions

When multiplying fractions, understanding the basic process is key. Whether the fractions are positive, negative, proper, or improper, the approach remains consistent.
First, identify the numerators and denominators of the fractions involved. If multiplying a whole number by a fraction, treat the whole number as a fraction with a denominator of one. For example, with \(-8\), treat it as \(-\frac{8}{1}\).
The multiplication typically follows these steps:

• Multiply the numerators of the fractions together.
• Multiply the denominators of the fractions together.

Finally, simplify the resulting fraction if needed. In some situations, simplifying involves finding the greatest common divisor (GCD) between the numerator and denominator. This makes the fraction easier to interpret and work with in further calculations.

Negative Numbers

Negative numbers can introduce complexity into operations if you're not careful. If you remember this simple rule, it will always guide you right: Multiplying a negative number by a positive number results in a negative number. This is because you are essentially taking the opposite or inverse of your positive result.
In our exercise, multiplying \(-8\) by \(\frac{1}{3}\) gives \(\frac{-8}{3}\). Here's a quick check:

• Negative \(-8\) simply means '8 less than zero.'
• \(\frac{1}{3}\) is just a proportion indicating a third of something.
• Carrying out the operation, your outcome is a portion of \(-8\), which gives \(\frac{-8}{3}\).

This outcome maintains the negative sign of \(-8\) despite the fraction being a positive influence.

Improper Fractions

Improper fractions occur when the numerator (top number) is larger than the denominator (bottom number), expressing a value greater than one. After multiplication, results often end as improper fractions in mathematics. In this exercise, the result \(\frac{-8}{3}\) is an improper fraction.
Working with improper fractions involves some choices:

• You can leave the result as it is, which is often acceptable in math.
• You can convert it to a mixed number. For \(\frac{-8}{3}\), this would result in \-2\frac{2}{3}\.

Both forms are correct, but choosing the right form depends on the context. In applications or tests, converting to mixed numbers can sometimes make the information more understandable.