Question

If \(4 a=3 b\) and \(6 b=0\), then: (A) \(\mathrm{a}=3 / 4\) (B) \(a=b\) (C) \(a=3\) and \(b=4\) (D) \(a / b=4 / 3\) (E) \(b / a=3 / 4\)

Short answer

None of the above

Step-by-step solution

Understand the given equations

The problem provides two equations: 1. \(4a = 3b\)2. \(6b = 0\)

Solve for b from the second equation

From the equation \(6b = 0\), we solve for \(b\):\[ 6b = 0 \implies b = 0 \]

Substitute b into the first equation

Now substitute \(b = 0\) into the first equation \(4a = 3b\):\[ 4a = 3(0) \implies 4a = 0 \]

Solve for a from the substituted equation

Solve for \(a\) from the equation \(4a = 0\):\[ 4a = 0 \implies a = 0 \]

Compare the solution to the given options

We found that \(a = 0\) and \(b = 0\). None of the provided options directly state this combination. However, we need to check the relationship between \(a\) and \(b\):\[ \frac{a}{b} =\] which is undefined since dividing by zero is undefined. Therefore, None of the given options are correct in strictest sense.So choose 'None of the above' as the final answer

Key concepts

equation solving

Equation solving can be understood as the process of finding the values (known as solutions) that satisfy the given equations. In this exercise, we were given two equations: 4a = 3b 6b = 0 The step-by-step solution involves a series of logical steps:

• First, solve for one variable using one of the equations. In this case, solving for b from 6b = 0 yields b = 0.
• Substitute the value of the solved variable back into the other equation. Substituting b = 0 into 4a = 3b simplifies as 4a = 0.
• Solve for the remaining variable. Solving 4a = 0 gives a = 0.

These steps are crucial to understand as they form the foundation for solving most types of algebraic equations.

algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It helps us solve equations like the ones given in the exercise. In algebra, mathematical symbols (like a and b) represent numbers in equations. Understanding basic algebraic principles is essential for solving equations:

• **Equality:** Understanding that both sides of an equation are equal is key. This allows us to apply operations to both sides of the equation.
• **Substitution:** This involves replacing a variable in one equation with its value from another equation, enabling us to solve more complex problems.
• **Isolation:** This technique involves manipulating the equation to isolate the variable we are solving for.

By understanding these basic principles, solving algebraic equations becomes straightforward and methodical.

GMAT preparation

Preparing for the GMAT requires a solid understanding of various mathematical concepts, including algebra and equation solving. Here are some tips to prepare effectively:

• **Practice Regularly:** Regular problem solving helps in identifying your strengths and weaknesses.
• **Understand Core Concepts:** Rather than just memorizing formulas, focus on understanding the underlying principles.
• **Simulate Test Conditions:** Practice with time limits to simulate actual test conditions as this helps in managing time better during the exam.
• **Review Mistakes:** Identify where you go wrong and work on those areas specifically.

By incorporating these tips into your preparation strategy, you will enhance your problem-solving skills and improve your GMAT score.

mathematical relationships

Understanding mathematical relationships is critical for solving problems on the GMAT. In this exercise, the relationship between a and b was given as 4a = 3b and 6b = 0. These relationships help us to:

• **Formulate Equations:** Translate word problems into mathematical equations.
• **Analyze Relationships:** Understand how different quantities relate to one another.
• **Solve Problems:** Use the relationships to find unknown values systematically.

By mastering these relationships, you can approach complex problems more logically and arrive at solutions more efficiently.