Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$-\frac{5}{8}+\frac{3}{4}$$

Step-by-Step Solution

Answer

The sum of -5/8 and 3/4 is 1/8.

Step 1: Identify the Common Denominator

First, check if the denominators of both fractions are the same. In this case, the denominators are 8 and 4. They are different, so find the common denominator. The least common denominator of 8 and 4 is 8.

Step 2: Rewrite the Fractions

Next, rewrite the fractions with the common denominator. The fraction -5/8 remains the same, because its denominator is already 8. Multiply the fraction 3/4 by 2/2 to make its denominator 8, and it becomes 6/8.

Step 3: Add the Fractions

Now, add the fractions -5/8 and 6/8. To do this, only add the numerators while the denominator remains 8: -5+6 = 1. So, the result is 1/8.

Step 4: Simplify the Result (if needed)

The last step is to simplify the result, if possible. But in this case, 1/8 is already in its simplest form. So, the sum of -5/8 and 3/4 is 1/8.

Key Concepts

Common Denominator

When you're adding fractions like \(-\frac{5}{8}\) and \(\frac{3}{4}\), having a common denominator is crucial. Denominators are the bottom numbers of fractions and represent how many equal parts the whole is divided into. Suppose the denominators are different, like in our example (8 and 4). In that case, you need to find a common ground, known as a "common denominator." In simple terms, a common denominator is a shared multiple of the denominators you have.
Why do we need it? It's like trying to add apples and oranges unless they have a common term, you can't directly add them together!
To determine a common denominator, find the **Least Common Multiple (LCM)** of the denominator values. In our case, the LCM of 8 and 4 is 8. This is because 8 is the smallest number that both 8 and 4 can divide into without a remainder.

• 8 ÷ 8 = 1
• 8 ÷ 4 = 2

This means 8 is the number we're looking for, turning our fractions into friends for easy addition.

Simplifying Fractions

When working with fractions, simplifying them is a necessary skill and involves reducing the fraction to its simplest form. A fraction is in its simplest form when the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1.
Simplifying can make fractions look cleaner and easier to handle, but it's possible only if the numerator and denominator share any common divisors.
Let's suppose you end up with a result like \(\frac{2}{8}\). Simplifying this fraction would involve dividing both the top and bottom by their greatest common divisor (GCD), which is 2:

• \(\frac{2}{8} = \frac{2 \div 2}{8 \div 2} = \frac{1}{4}\)

However, in our original problem, the result \(\frac{1}{8}\) is already simplified because 1 and 8 have no common divisors except for 1. Therefore, no further simplification is required, making it the simplest form already!

Numerators and Denominators

Understanding the roles of numerators and denominators in fractions is fundamental to mastering fraction operations. The **numerator** is the top part of a fraction and indicates how many parts we are referring to, while the **denominator** is the bottom part, showing into how many equal parts the whole is divided. In fraction \(\frac{a}{b}\), \(a\) is the numerator and \(b\) is the denominator.
When adding fractions like \(-\frac{5}{8}\) and \(\frac{6}{8}\), focus on the numerators since the denominators are already common. Simply add or subtract the numerators, according to the operation indicated:

• \(-5 + 6 = 1\)

The denominator stays the same because it represents the same unit size or part of the whole. In our solved problem, this approach helped us sum the fractions up easily to \(\frac{1}{8}\).
To be skillful with fractions, always watch both the numerator and denominator! They guide how fractions interact with each other in addition, subtraction, and other processes.