Problem 79 - Prealgebra

Question

Combine. \(\frac{3}{4}+\frac{5}{8}\)

Step-by-Step Solution

Answer

\(\frac{3}{4} + \frac{5}{8} = \frac{11}{8}\)

Step 1: Find a Common Denominator

To add fractions, the denominators must be the same. The denominators here are 4 and 8. The least common multiple of 4 and 8 is 8. Therefore, 8 will be our common denominator.

Step 2: Convert Fractions to Equivalent Fractions with Common Denominator

Convert \(\frac{3}{4}\) to an equivalent fraction with a denominator of 8. Multiply both the numerator and denominator by 2: \(\frac{3 \times 2}{4 \times 2} = \frac{6}{8}\). The fraction \(\frac{5}{8}\) already has a denominator of 8, so no conversion is needed.

Step 3: Add the Numerators

Now that the denominators are the same, we add the numerators of the equivalent fractions: \(6 + 5 = 11\).

Step 4: Write the Result as a Single Fraction

The result of adding the fractions is the sum of the numerators over the common denominator. So, \(\frac{6}{8} + \frac{5}{8} = \frac{11}{8}\).

Key Concepts

Understanding a Common Denominator

In the realm of fractions, to perform operations like addition or subtraction, we need a common denominator. This is because fractions with different denominators represent parts of wholes that are divided differently. To add these fractions, they need to be speaking the same "language." You can think of a common denominator as a type of "translator" that allows fractions to communicate effectively. Finding a common denominator involves identifying a number that both denominators can divide into evenly. This simplifies the addition process and ensures accuracy.

• For example, when working with fractions like \( \frac{3}{4} \) and \( \frac{5}{8} \), we need a common denominator for them to meet on the same ground.
• Here, the common denominator is 8 because 8 is a multiple of both 4 and 8.

Without finding a common denominator, you'll struggle to add fractions correctly. It acts as the foundational step to simplify the addition process, avoiding errors in calculation.

Least Common Multiple Simplifies Fraction Tasks

The Least Common Multiple (LCM) is a vital tool in fraction operations, especially when finding a common denominator. The LCM of two numbers is the smallest number that both can divide into without leaving a remainder. When it comes to adding fractions, like in our example of \( \frac{3}{4} \) and \( \frac{5}{8} \), the LCM efficiently provides the smallest usable denominator. Often, finding the LCM simplifies calculations and ensures you're dealing with the smallest numbers possible, reducing the chances of mistakes.

Importance of LCM

• It minimizes the numbers we work with, making calculations easier and quicker.
• Ensures that your solution is simplified as much as possible from the start.
• Creates a direct path to finding equivalent fractions with ease.

Establishing the LCM early on helps set up the parameters for smooth operations when working with fractions, simplifying both multiplication and division tasks.

The Role of Equivalent Fractions

Equivalent fractions are crucial when working with fractions and performing operations like addition or subtraction. These fractions, though they may appear different, actually represent the same value. They are essentially different expressions of the same part of a whole. When converting to equivalent fractions, you're ensuring that all parts can be uniformly compared and calculated. For example, converting \( \frac{3}{4} \) into \( \frac{6}{8} \) involved multiplying both the numerator and denominator by the same number, 2. This process doesn't change the value, merely the way it's expressed, aligning \( \frac{3}{4} \) with a denominator of 8, compatible for addition with \( \frac{5}{8} \).

How do Equivalent Fractions work?

• They provide a common basis for arithmetic operations.
• Make it easier to visualize and manage fractions together.
• Ensure consistency in mathematical expressions and comparisons.

Equivalent fractions act as the stepping stone for adding, subtracting, and even simplifying fractions, giving you the flexibility to bring different parts into a unified whole for calculation.