Problem 37 - Prealgebra

Question

Find each sum or difference. Write in simplest form. \(\frac{x}{8}+\frac{4 x}{8}\)

Step-by-Step Solution

Answer

The sum is \(\frac{5x}{8}\).

Step 1: Identify Like Terms

We notice that both terms in the given expression, \(\frac{x}{8}\) and \(\frac{4x}{8}\), share the same denominator, 8. This indicates that they're like terms and can be combined directly.

Step 2: Combine the Numerators

Since the denominators are the same, we can add the numerators. Thus, the expression becomes \(\frac{x + 4x}{8}\).

Step 3: Simplify the Numerator

Combine the terms in the numerator: \(x + 4x = 5x\). Therefore, the expression simplifies to \(\frac{5x}{8}\).

Step 4: Write in Simplest Form

Since \(\frac{5x}{8}\) is already in its simplest form (as 5 and 8 have no common factors other than 1), the simplest form of the expression is \(\frac{5x}{8}\).

Key Concepts

Fractions

Fractions are mathematical expressions representing the division of one quantity by another. They consist of a numerator (the top number) and a denominator (the bottom number). Fractions can be tricky at first, but once you get the hang of them, they become a powerful tool for solving problems. In an expression like \(\frac{x}{8}\), \(x\) is the numerator and 8 is the denominator. This means \(x\) is divided by 8.

When dealing with fractions, it's essential to pay attention to the denominators, especially when performing operations like addition or subtraction. Only fractions with the same denominator (also known as 'like denominators') can be directly added or subtracted. If the denominators are not the same, we'll need to find a common denominator first or use equivalent fractions.

Remember:

• Like terms with the same denominator make it easy to perform addition or subtraction.
• Always ensure that the fraction is in its simplest form after calculations.
• Fractions can represent parts of whole numbers, ratios, or even unknown values such as variables.

With a bit of practice, you'll soon navigate fractions with confidence!

Like Terms

In algebra, 'like terms' refer to terms that have the exact same variable parts. For example, in the expression \(\frac{x}{8} + \frac{4x}{8}\), both terms are like because they both include the variable \(x\) and they share the same denominator. Like terms can be combined, making algebraic expressions simpler and easier to solve.

Here's why like terms matter:

• They allow for direct addition or subtraction without needing any additional steps.
• Combining them simplifies expressions, reducing the number of terms.
• They help maintain the clarity of expressions, as combining them highlights the common factors.

For instance, combining \(x\) and \(4x\) gives \(5x\). This doesn’t change the fundamental nature of the expression, but makes it more straightforward, showing how the terms relate directly. Remember, only coefficients (the numbers in front of the variables) are manipulated when combining like terms.

Simplifying Expressions

Simplifying expressions is about making them as straightforward and compact as possible. It involves combining like terms, reducing coefficients where needed, and ensuring the final expression is clear and understandable.

Let's take the expression \(\frac{x + 4x}{8}\) as an example. By combining the terms in the numerator, you obtain \(5x\). This gives you \(\frac{5x}{8}\).

The goal of simplification is:

• To make the expression easier to work with.
• To reveal any underlying relationships or factors in the expression.
• To make subsequent operations, like further arithmetic or algebraic manipulation, more manageable.

It's important to reduce the expression to its basic form without changing its value. When reducing fractions, check for common factors in the numerator and the denominator. If there are none, the fraction is already in its simplest form.

Simplification keeps equations neat and comprehensible, making problem-solving both efficient and effective.