Question

Quantity_A \(4\left(\frac{1}{2} x+2 y\right)\) Quantity \(\mathbf{B}\) \(2 x+8 y\) A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

Short answer

The two quantities are equal, hence the correct answer is C. The two quantities are equal.

Step-by-step solution

Simplify Quantity A

To simplify Quantity A, distribute the 4 inside the parentheses to both \( \frac{1}{2} x\) and \(2y\). This gives us \(2x + 8y\).

Compare Quantity A and Quantity B

Now, let's compare the simplified expressions for Quantity A and Quantity B. Seeing that both quantities are \(2x + 8y\), we can say that Quantity A is equal to Quantity B.

Key concepts

Distributive Property

When you encounter an expression like \( 4\left(\frac{1}{2}x + 2y\right) \), the distributive property allows you to simplify it by multiplying each term inside the parentheses by the factor outside. In this case, the number outside the parentheses is 4. This technique helps to eliminate the parentheses and simplifies calculations.
To apply the distributive property:

• Multiply the 4 by \( \frac{1}{2}x \), resulting in \( 2x \). This is because \(4 \times \frac{1}{2} = 2\).
• Then, multiply the 4 by \( 2y \), resulting in \( 8y \). This is calculated as \(4 \times 2 = 8\).

Using these steps, the expression \( 4\left(\frac{1}{2}x + 2y\right) \) can be simplified to \( 2x + 8y \).
This simplification is crucial for comparing expressions, as it offers a clear, straightforward form that can be easily evaluated alongside another expression.

Expressions Comparison

Once you've simplified expressions, comparing them becomes straightforward. In problems like the one given, you are asked to determine the relationship between two quantities: Quantity A and Quantity B. The aim is to see if they are equal, or if one is larger than the other.
After using the distributive property, both Quantity A (\( 4\left(\frac{1}{2}x + 2y\right) \)) and Quantity B (\( 2x + 8y \)) simplify to \( 2x + 8y \). This indicates both expressions are identical.
When comparing expressions:

• Check if the simplified forms are equivalent. In this case, \( 2x + 8y \) is the same for both quantities.
• If the expressions are equivalent, conclude that they are equal, meaning neither is greater, nor less than the other.

This form of expression comparison is fundamental in quantitative reasoning, particularly in standardized tests like the GRE.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves making them more concise without changing their value. It is a key step in solving and comparing mathematical problems. In the problem given, we already applied the distributive property to simplify Quantity A from \( 4\left(\frac{1}{2}x + 2y\right) \) to \( 2x + 8y \).
Here's why simplification is important:

• It helps in identifying equivalent expressions, which simplifies the problem-solving process.
• A simplified expression is easier to interpret and compare, crucial for recognizing whether one quantity may be greater than, less than, or equal to another.

By understanding how to simplify expressions, you gain an advantage in tackling complex problems effortlessly. Removing brackets and combining like terms make expressions cleaner and more manageable, paving the way for an accurate analysis of their relationships.