Problem 68 - Prealgebra

Question

Apply the distributive property, then find the LCD and simplify. $$\frac{2}{3} x+\frac{5}{6} x$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3}{2}x \).

Step 1: Apply the Distributive Property

The problem given is \(\frac{2}{3}x + \frac{5}{6}x\). We can factor out the common variable \(x\) from both terms, giving us \(x \left( \frac{2}{3} + \frac{5}{6} \right)\).

Step 2: Find the Least Common Denominator (LCD)

To add the fractions \( \frac{2}{3} \) and \( \frac{5}{6} \), we must find a common denominator. The least common denominator of 3 and 6 is 6.

Step 3: Rewrite Fractions with the LCD

Rewrite \( \frac{2}{3} \) with a denominator of 6: \( \frac{2}{3} = \frac{4}{6} \), because \(2 \times 2 = 4\) and \(3 \times 2 = 6\). The second fraction already has a denominator of 6, so it remains \( \frac{5}{6} \).

Step 4: Add the Fractions

Now add the fractions: \( \frac{4}{6} + \frac{5}{6} = \frac{9}{6} \).

Step 5: Simplify the Fraction

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

Step 6: Write the Final Expression

Substitute back into the expression from Step 1: \(x \left( \frac{3}{2} \right)\), which is equal to \( \frac{3}{2}x\).

Key Concepts

Factoring Expressions

Factoring is an essential tool in algebra that simplifies expressions. Imagine it like breaking down a group into smaller, identical parts to manage better. When you have an expression with a common variable across terms, you can 'factor' this variable out. Let's look at how this works: given an expression like \( \frac{2}{3} x + \frac{5}{6} x \), notice the common factor, \( x \). Factoring this out means you are essentially saying, "If each part of my expression has an \( x \), let's just deal with the numbers first and reattach \( x \) later." Thus, you can rewrite it as \( x \left( \frac{2}{3} + \frac{5}{6} \right) \). This simplification makes it easier to manage complex expressions, especially when combined with tasks like finding a least common denominator or simplifying fractions. Factoring is the first step in cleaning up your math problems!

Least Common Denominator

A common issue in handling fractions involves finding a denominator that works consistently across all terms, known as the least common denominator (LCD). The LCD is fundamental when you're adding or subtracting fractions because it allows you to convert different denominators into a shared one, simplifying the process. Take our example, \( \frac{2}{3} + \frac{5}{6} \). Here, the denominators are 3 and 6. The simplest common number between them that they can both divide into evenly is 6. Here's how to use the LCD:

• Identify each fraction's denominator, in this case, 3 and 6.
• Determine the smallest number both denominators can divide without a remainder. This is 6 for our fractions.
• Adjust the fractions so both have the denominator 6, turning \( \frac{2}{3} \) into \( \frac{4}{6} \) by multiplying the numerator and denominator by 2.

By finding and using the LCD, combining fractions becomes as easy as adding whole numbers!

Simplifying Fractions

Simplifying fractions is like reducing a recipe to its most efficient form; you want to cut it down to its lowest terms without changing its value. This involves finding the greatest common divisor (GCD) of the numerator and denominator, and then dividing them both by this number. In our progression, we reached the fraction \( \frac{9}{6} \). The GCD of 9 and 6 is 3. By dividing them both by 3, we simplify \( \frac{9}{6} \) to \( \frac{3}{2} \). Here's a simpler process:

• Determine the greatest number that divides both your numerator and denominator evenly.
• Divide both the numerator and denominator by this number to achieve the simplest form.

Doing this reduces complexity, making calculations and further operations more manageable. It's a bit like clearing the clutter to see clearly whatever you're working on. Simplifying makes fractions neat, tidy, and less likely to trip you up in later steps!