Problem 66 - Algebra A Combined Function

Question

Perform each indicated operation. See Section R .2. $$ \frac{4}{3}-\frac{8}{3} $$

Step-by-Step Solution

Answer

The result is \(-\frac{4}{3}\).

Step 1: Identify the Problem Type

The problem asks us to perform the operation on two fractions. Since both fractions have the same denominator, this is a subtraction of fractions with like denominators.

Step 2: Subtract the Numerators

When subtracting fractions with the same denominator, subtract the numerators and keep the denominator the same. Here, subtract the numerator 8 from 4: \(4 - 8 = -4\).

Step 3: Retain the Common Denominator

Since both fractions have the same denominator (3), the result will also have this denominator. Thus, the fraction is:\(-\frac{4}{3}\).

Step 4: Simplify the Result (if needed)

Since \(-\frac{4}{3}\) is already in simplest form, no further simplification is needed.

Key Concepts

Fractions with Like Denominators

When subtracting fractions that share the same denominator, the process becomes straightforward. This is known as subtracting fractions with like denominators. Imagine both fractions as slices of the same pie. Since the size of the slices (the denominators) is the same, we only need to focus on how many slices there are (the numerators).

In the exercise given, both fractions have a denominator of 3. Therefore, you only need to subtract the numerators, which are 4 and 8 in this case. Retain the common denominator of 3 to form the resulting fraction.

• Step 1: Identify like denominators.
• Step 2: Subtract the numerators.
• Step 3: Keep the same denominator.

Following these steps ensures you correctly subtract fractions without any hassle.

Simplifying Fractions

Simplifying fractions is all about making the fraction as simple as possible. This means reducing the numerator and denominator to the smallest possible values while retaining the same value of the fraction. However, not all fractions can be simplified. Some are already in their simplest form, which means you can't reduce them any further without changing their value.

To simplify a fraction, you find the greatest common divisor (GCD) of both the numerator and the denominator and divide both by this number. In the exercise, the fraction \(-\frac{4}{3}\) is already in its simplest form, because 4 and 3 do not share any common factors other than 1. This means no further division is possible. Here are the essential steps if simplification is required:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Reconstruct the fraction with these simplified numbers.

Understanding when and how to simplify fractions can make solving problems much easier.

Numerators and Denominators

Understanding numerators and denominators is key to mastering fractions. The numerator is the number above the line in a fraction, representing the parts you have. If you picture a pizza, the numerator is how many slices are eaten or remaining. The denominator, below the line, shows the total number of equal parts in the whole or the pizza.

In the subtraction process, it's the numerators that interact while the denominator stays constant if they are "like" denominators. In our example, where we subtract \(\frac{8}{3}\) from \(\frac{4}{3}\), the numerators are subtracted: 4 - 8 = -4. The denominator, 3, is the same and does not change.

• Numerator: Top part, tells how many parts.
• Denominator: Bottom part, tells size of each part.
• In like denominators, only numerators change in operations.

By understanding the role each part plays, you can perform operations on fractions more accurately and confidently.