Problem 348 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ \frac{1}{5}(4 m+20) $$

Step-by-Step Solution

Answer

\( \frac{4m}{5} + 4 \)

Step 1: Distribute the Fraction

Apply the Distributive Property by multiplying the fraction \(\frac{1}{5}\) with each term inside the parentheses. So, multiply \( \frac{1}{5} \times 4m \) and \( \frac{1}{5} \times 20 \).

Step 2: Simplify Each Term

Calculate each multiplication separately. First, \( \frac{1}{5} \times 4m = \frac{4m}{5} \). Then, \( \frac{1}{5} \times 20 = 4 \).

Step 3: Combine the Results

Add the simplified terms together: \( \frac{4m}{5} + 4 \). This is the simplified form of the given expression.

Key Concepts

simplifying expressions

Simplifying expressions is a fundamental concept in algebra. It means making an expression as simple as possible. You do this by combining like terms and performing basic arithmetic operations.
Let's take the example from the exercise. We started with \(\frac{1}{5}(4m + 20)\).
The goal was to simplify this expression using the Distributive Property, which we'll cover in more detail later.
Once you apply the distributive property, you need to handle each term individually. Here's where you simplify them. For instance, in the step-by-step solution, \( \frac{1}{5} \times 4m = \frac{4m}{5} \) is simplified by multiplying the fraction with the term directly. Similarly, \( \frac{1}{5} \times 20 = 4 \) simplifies the fraction multiplied by the number.
When simplifying expressions, always remember to:

• Combine like terms
• Factor where possible
• Reduce fractions properly

This will help you make the algebraic expression much simpler and easier to handle in more complex math problems.

fractions in algebra

Fractions can sometimes seem tricky when they pop up in algebra. But, don't worry, understanding a few basic principles can make it much easier.
Fractions consist of a numerator (the top number) and a denominator (the bottom number). When you multiply a fraction by another number, you multiply the numerator by that number and keep the denominator the same. For example, in the given exercise, we had to multiply \( \frac{1}{5} \) with each term inside the parentheses.
Let's recap the steps:

• Multiply \( \frac{1}{5} \) by \( 4m \), yielding \( \frac{4m}{5} \).
• Multiply \( \frac{1}{5} \) by \( 20 \), yielding \( 4 \).

These fractions are simplified correctly. Remember, multiplying fractions doesn't change the denominator, only the numerator. Simplifying fractions in algebra is a basic skill that will assist you greatly in more advanced problems.

distributive property

The Distributive Property is a key algebraic tool. It states that \( a(b + c) = ab + ac \). In other words, multiplying a sum by a number gives the same result as multiplying each addend individually by the number and then adding the products.
In our exercise, we used the Distributive Property to simplify the expression \( \frac{1}{5}(4m+ 20) \).
This meant:

• Multiplying \( \frac{1}{5} \) by \( 4m \) to get \( \frac{4m}{5} \)
• Multiplying \( \frac{1}{5} \) by \( 20 \) to get \( 4 \)

Then, we combined these results to get the final simplified expression: \( \frac{4m}{5} + 4 \).
The Distributive Property helps you expand and simplify expressions, especially when dealing with parentheses. Ensure to apply this property correctly for precise and accurate solutions in algebra.