Problem 47 - Prealgebra

Question

The problems below review some basic concepts of addition of fractions and mixed numbers. Add each of the following and reduce all answers to lowest terms. $$\frac{3}{4}+\frac{2}{3}$$

Step-by-Step Solution

Answer

The sum is \(1 \frac{5}{12}\).

Step 1: Find a Common Denominator

To add fractions, we need a common denominator. For \( \frac{3}{4} \) and \( \frac{2}{3} \), the least common multiple of 4 and 3 is 12. So, we will use 12 as the common denominator.

Step 2: Convert Fractions to Equivalent Fractions

We convert both fractions to have the common denominator 12. For \( \frac{3}{4} \), multiply the numerator and denominator by 3 to get \( \frac{9}{12} \). For \( \frac{2}{3} \), multiply the numerator and denominator by 4 to get \( \frac{8}{12} \).

Step 3: Add the Fractions

Now that both fractions have the same denominator, we can add them: \[ \frac{9}{12} + \frac{8}{12} = \frac{17}{12}. \]

Step 4: Simplify the Result

The result \( \frac{17}{12} \) is an improper fraction. We can also express it as a mixed number: 17 divided by 12 equals 1 with a remainder of 5, so it can be written as \(1 \frac{5}{12}\).

Key Concepts

Understanding Fractions

Fractions are numerical values that represent parts of a whole. They consist of two main components: the numerator and the denominator. The numerator, found at the top, tells you how many pieces you have, while the denominator, found at the bottom, tells you how many equal pieces make up the whole.
Fractions can be less than, equal to, or greater than 1. When the numerator is smaller than the denominator, the fraction is proper and less than 1. When the numerator is greater than the denominator, it is improper and can be greater than 1. Understanding this is key when adding fractions or converting improper fractions to mixed numbers.
When working with fractions, always strive to simplify them by dividing both the numerator and the denominator by their greatest common factor. This makes calculations easier and results more visually clean. Simplifying fractions doesn’t change the amount; it just expresses it in its simplest form.

The Importance of a Common Denominator

A common denominator is crucial when adding fractions. It refers to a shared multiple of the original denominators in the fractions you are working with. This shared denominator allows you to add fractions directly as if they had the same base.
Finding the least common denominator often involves determining the least common multiple (LCM) of the denominators involved. For instance, when adding fractions like \(\frac{3}{4}\) and \(\frac{2}{3}\), the LCM of 4 and 3 is 12.

• Find the least common multiple of the denominators.
• Convert both fractions so that they share this denominator.

By converting fractions to this common basis, it levels the playing field, allowing them to be summed accurately without altering their original values.

Converting and Equivalency in Fractions

To add fractions, converting them to have the same denominator is necessary to achieve equivalency. This step involves using multiplication to adjust each fraction’s numerator and denominator while keeping their proportional sizes the same.
Take \(\frac{3}{4}\). To convert it to have the common denominator of 12, multiply both the top and bottom by 3, resulting in \(\frac{9}{12}\). Similarly, for \(\frac{2}{3}\), multiply the numerator and denominator by 4 to get \(\frac{8}{12}\).

• Identify the conversion factor for each fraction.
• Multiply both numerator and denominator by this factor.

Converting fractions to equivalent forms allows them to fit neatly onto a shared timeline, simplifying operations such as addition or subtraction.

Dealing With Mixed Numbers

Mixed numbers are a combination of whole numbers and proper fractions, used when improper fractions arise. They help in visualizing amounts greater than 1, making them easier to comprehend and use in calculations.
For example, the improper fraction \(\frac{17}{12}\) can also be expressed as the mixed number \(1\frac{5}{12}\). This is done by dividing the numerator by the denominator: 17 divided by 12 is 1, with a remainder of 5.

• Divide the numerator by the denominator.
• Write down the whole number result.
• The remainder becomes the new numerator over the same denominator.

Mixed numbers are practical in various scenarios, especially when needing a more intuitive depiction of an amount that isn’t entirely fractional or whole.