Problem 76 - Prealgebra

Question

Apply the distributive property, then find the LCD and simplify. $$\frac{3 x}{5}-\frac{3 x}{8}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{9x}{40} \).

Step 1: Identify the Need for a Common Denominator

We have two fractions: \( \frac{3x}{5} \) and \( \frac{3x}{8} \). To apply arithmetic operations, we need a common denominator. This requires finding the least common multiple of the denominators 5 and 8.

Step 2: Find the Least Common Denominator (LCD)

The denominators are 5 and 8. The least common multiple of these numbers is 40, which is the smallest number that both 5 and 8 divide evenly.

Step 3: Rewrite Each Fraction with the LCD

Convert each fraction to have a denominator of 40. For \( \frac{3x}{5} \), multiply the numerator and denominator by 8 to get \( \frac{3x imes 8}{5 imes 8} = \frac{24x}{40} \). For \( \frac{3x}{8} \), multiply the numerator and denominator by 5 to get \( \frac{3x imes 5}{8 imes 5} = \frac{15x}{40} \).

Step 4: Subtract the Fractions

Now both fractions have the same denominator, 40, so they can be subtracted directly. Subtract the numerators: \( \frac{24x}{40} - \frac{15x}{40} = \frac{24x - 15x}{40} = \frac{9x}{40} \).

Step 5: Simplify if Necessary

The fraction \( \frac{9x}{40} \) is already in its simplest form, so no additional simplification is necessary.

Key Concepts

Distributive Property

The distributive property is a helpful algebraic concept that shows how to multiply a single term and two or more terms inside parentheses. It's like distributing whatever is outside the parentheses to each term inside. Here’s a quick breakdown: If we have an expression like

• \( a(b + c) = ab + ac \),

the distributive property tells us to multiply \( a \) by both \( b \) and \( c \), giving us the expanded form. This property is really handy when working with fractions and equations, enabling us to simplify and rearrange parts of an expression for easier calculation. Though the initial problem focuses on applying the distributive property to settings rather than directly, this mental model is essential when transitioning between expanded and factored forms while working through math problems.

Fractions

Fractions can seem tricky at first, but they are just a way of representing parts of a whole. A fraction consists of two parts: a numerator (the top number) indicating how many parts we have, and a denominator (the bottom number) indicating how many equal parts the whole is divided into. Understanding fractions is crucial because they appear everywhere in mathematics, from dividing whole numbers to working with variables like in our given exercise. In the example

• \( \frac{3x}{5} \),

the numerator \( 3x \) indicates the quantity, while the denominator 5 tells us how many parts the whole is divided into. To compute operations with fractions, especially adding or subtracting them, it is important to have common denominators, which leads us to the concept of the least common denominator.

Simplification

Simplification in mathematics means making an expression or equation as simple as possible without changing its value. This process is essential for easier calculations and better understanding of mathematical expressions. When simplifying fractions, the goal is to reduce them to their simplest form by dividing the numerator and the denominator by their greatest common factor. In our example, after subtracting the two fractions and obtaining

• \( \frac{9x}{40} \),

we would check if there’s a common factor other than 1 that can divide both the numerator and the denominator. If not, the fraction is already simplified. Simplification isn't always about changing fractions—sometimes, it’s about ensuring that your mathematical expression is manageable and easy to understand, which is the heart of making math approachable and less intimidating.