Question

If \(5 a=2 b\) and \(6 b=12\), then: (A) \(a / b=2 / 5\) (B) \(a=b / 4\) (C) \(a=5\) and \(b=2\) (D) \(a / b=5 / 2\) (E) \(b / a=10 / 6\)

Short answer

(A) \( a / b = 2 / 5 \)

Step-by-step solution

- Solve for b

Given the equation: \[6b = 12\] Divide both sides by 6 to find the value of b: \[b = \frac{12}{6} = 2\]

- Solve for a

Substitute the value of b into the equation: \[5a = 2b\] Now, replace b with 2: \[5a = 2 \cdot 2\] Simplify to find the value of a: \[5a = 4\] Divide both sides by 5: \[a = \frac{4}{5}\]

- Compare the Ratios

With \(a\) and \(b\) known, compare \(\frac{a}{b}\) and \(\frac{b}{a}\): \[\frac{a}{b} = \frac{\frac{4}{5}}{2} = \frac{4}{5} \cdot \frac{1}{2} = \frac{4}{10} = \frac{2}{5}\] This matches option (A).

Key concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. In this exercise, algebra helps us solve for variables by performing operations and manipulating the given equations. For example, starting from the equation \[5a = 2b\], we can solve for individual variables like \(a\) and \(b\). Here, we need to isolate \(b\) first from \[6b = 12\]. By dividing both sides of the equation by 6, we find that \(b = 2\).

Next, substituting \(b\) into \[5a = 2b\] lets us solve for \(a\). Replacing \(b\) with 2, we get \[5a = 2 \cdot 2\]. Simplifying further, we find that \[5a = 4\]. Finally, dividing both sides by 5 gives us \(a = \frac{4}{5}\).

Algebra provides powerful tools for solving equations systematically, making it easier to find relationships between variables.

Ratios

Ratios are a way to compare two quantities by expressing how many times one value contains or is contained within the other. They are often represented in the form of fractions. In the GMAT problem, the goal is to compare the variables \(a\) and \(b\) using ratios. Once we know the values of \(a\) and \(b\), which are \(\frac{4}{5}\) and 2, respectively, we can compare them as follows:

To find the ratio\(\frac{a}{b}\):
\[\frac{a}{b} = \frac{\frac{4}{5}}{2} = \frac{4}{5} \cdot \frac{1}{2} = \frac{4}{10} = \frac{2}{5}\]
Ratios can help you understand the proportion or relationship between different quantities. In this case, \[\frac{a}{b} = \frac{2}{5}\] shows the proportion of \(a\) to \(b\) clearly.

Graduate Management Admission Test

The Graduate Management Admission Test (GMAT) is a standardized exam used globally by business schools to assess the qualifications of applicants for advanced study in business and management. Problem-solving questions like this one, involving algebra and ratios, are common on the GMAT.

Mastering these concepts helps you in multiple areas of the test. For instance, questions often require quick manipulation of equations, an understanding of proportions, and the ability to compare different variables logically.

Practicing these types of problems familiarizes you with the test's format and types of questions, enhancing your ability to answer efficiently under timed conditions. The ability to analyze and break down problems into simpler steps is crucial for success on the GMAT.