Problem 38 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ -\frac{5}{6}+\left(-\frac{2}{3}\right) $$

Step-by-Step Solution

Answer

The result is \(-\frac{3}{2}\).

Step 1: Identify the Common Denominator

First, we need to identify a common denominator for the fractions \(-\frac{5}{6}\) and \(-\frac{2}{3}\). The denominators are 6 and 3, respectively. The least common denominator is 6.

Step 2: Convert Fractions

Convert \(-\frac{2}{3}\) into an equivalent fraction with a denominator of 6. Multiply both the numerator and the denominator by 2: \(-\frac{2}{3} = -\frac{4}{6}\). Now, we have two fractions: \(-\frac{5}{6}\) and \(-\frac{4}{6}\).

Step 3: Add the Fractions

Now that the fractions have the same denominator, add the numerators: \(-5 + (-4) = -9\). So, \(-\frac{5}{6} + (-\frac{4}{6}) = -\frac{9}{6}\).

Step 4: Simplify the Result

Simplify \(-\frac{9}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3: \(-\frac{9}{6} = -\frac{3}{2}\).

Key Concepts

Common Denominator

When adding fractions, a common denominator is crucial to ensure the fractions can be combined easily. This is because fractions with different denominators are like parts of a pie divided into different numbers of slices. To find a common denominator, look for the least common multiple (LCM) of the denominators involved. For example, in the original exercise with the fractions

• \(-\frac{5}{6}\)
• \(-\frac{2}{3}\)

the denominators are 6 and 3. The LCM of these numbers is 6. Choosing a common denominator means you can add the fractions directly, making calculations much simpler. When all fractions share the same denominator, you only need to add the numerators to find the sum of the fractions.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value or proportion. In the exercise, we converted \(-\frac{2}{3}\) to an equivalent fraction with a denominator of 6. To achieve this, multiply the numerator and the denominator by the same number. In this case, multiplying both by 2 gives us:

• \(-\frac{2}{3} = -\frac{4}{6}\)

This transformation does not change the value of the fraction; it only adjusts its form so that it can be easily added to other fractions with the same denominator. Finding equivalent fractions is a powerful tool for making fraction addition and subtraction much easier. It allows you to work with like terms, simplifying many mathematical processes.

Simplifying Fractions

After performing operations with fractions, it's often best to simplify the result. Simplifying a fraction involves reducing it to its simplest form so that both the numerator and the denominator have no common factors other than 1. For instance, after adding

• \(-\frac{5}{6}\)
• \(-\frac{4}{6}\)

we arrived at \(-\frac{9}{6}\). The greatest common divisor (GCD) of 9 and 6 is 3. By dividing both the numerator and the denominator by this number, the fraction is simplified to

• \(-\frac{3}{2}\)

A simplified fraction is not only easier to interpret but also makes further calculations less complex. It's an important final step in fraction work that ensures your answer is as clear and concise as possible.

Negative Numbers

Dealing with negative numbers can sometimes be tricky, especially in arithmetic operations such as addition. When adding negative fractions, it's important to follow the same rules you use for whole numbers. Negative numbers indicate a value below zero, and when adding them you are effectively moving further into the negative territory. In the exercise:

• \(-5\) and \(-4\) were added together to get \(-9\)

This principle works for fractions just as it does for whole numbers. Always combine the negative signs appropriately, being careful not to change the actual values you're working with. Understanding how to handle negatives correctly is necessary for achieving the right result when adding, subtracting, or performing any mathematical operation with them.