Question

When $x>0$, which of these expressions is equivalent to $\frac{1}{\frac{1}{2x}}+\frac{3}{\frac{6}{4x}}$ ? (A) $4x$ (B) $7x$ (C) $\frac{1}{2}x-4$ (D) $x^2-12$

Short answer

The expression is equivalent to $4x$, which is option (A).

Step-by-step solution

Simplify the first term

The first term of the expression is $\frac{1}{\frac{1}{2x}}$. Simplify this expression by taking the reciprocal. The reciprocal of $\frac{1}{2x}$ is $2x$. Thus, $\frac{1}{\frac{1}{2x}}=2x$.

Simplify the second term

The second term of the expression is $\frac{3}{\frac{6}{4x}}$. Simplify this expression by taking the reciprocal of the denominator: The reciprocal of $\frac{6}{4x}$ is $\frac{4x}{6}$. Therefore, $\frac{3}{\frac{6}{4x}}=3\times \frac{4x}{6}=\frac{12x}{6}=2x$.

Combine the simplified terms

After simplifying both terms, we have $2x$ and $2x$. Add these expressions together: $2x+2x=4x$.

Evaluate answer options

The simplified expression is $4x$. Compare this with the given answer choices: (A) $4x$, (B) $7x$, (C) $\frac{1}{2}x-4$, (D) $x^2-12$. The correct choice is (A) $4x$.

Key concepts

Exploring Reciprocals

In mathematics, the concept of a reciprocal is a fundamental tool for simplifying expressions and solving equations. A reciprocal of a number is simply one divided by that number.
For example, the reciprocal of a fraction like $\frac{1}{2x}$ is just flipping the fraction, giving us $2x$. Think of it like turning the fraction upside down.
When you have the reciprocal of any fraction $\frac{a}{b}$, it becomes $\frac{b}{a}$. This swapping of numerator and denominator is key in both simplifying complex fractions and solving algebraic expressions on standardized tests like the PSAT.
Understanding reciprocals helps in speeding up problem-solving, especially when dealing with multiple layers of fractions. It's a skill that comes in handy in many algebra problems.

Mastering Simplifying Expressions

Simplifying expressions is an important skill in algebra, often tested in exams like the PSAT. This process involves reducing an expression to its simplest form, which makes it easier to interpret and solve.
To simplify an expression:

• Identify terms that can be combined. This might involve similar variables or common denominators.
• Utilize the properties of operations, such as distributive, associative, and commutative properties.
• Apply operations like taking out the reciprocal as demonstrated in the problem to make clear fractions into more manageable terms.

Take the example $\frac{1}{\frac{1}{2x}}$. By finding the reciprocal and simplifying correctly, we reduce the expression to $2x$. For $\frac{3}{\frac{6}{4x}}$, the reciprocal simplifies it down eventually to $2x$, proving how powerful this technique is.
Through these steps, simplifying allows us to transform expressions into a form that is much easier to handle, particularly when answering multiple-choice questions.

Navigating PSAT Algebra Problems

Algebra problems on the PSAT often require a blend of different mathematical concepts. Key abilities tested include simplifying expressions, solving equations, understanding functions, and manipulating algebraic terms.
To excel in these questions:

• Keep solid basic skills in mind, such as understanding variables and coefficients.
• Improve proficiency with fractions and reciprocals, as they frequently appear in complex forms.
• Always practice simplifying expressions and solve as many practice problems as you can to build confidence.

Recall that in the presented exercise, recognizing the reciprocal and simplifying expressions allowed us to find the equivalent of $\frac{1}{\frac{1}{2x}}+\frac{3}{\frac{6}{4x}}$ as $4x$.
This problem-solving style, typical of PSAT algebra questions, involves more than just calculations; it requires logical reasoning and methodical simplification. Understanding the types of algebraic manipulations likely to be encountered allows students to tackle these questions with ease.