Question

If \(a / 2+b / 3=3\) and \(a-b=3\), what is \(a+b\) ? (A) \(5 / 3\) (B) 2 (C) 3 (D) \(2.5\) (E) \(6.6\)

Short answer

The value of \(a + b\) is 6.6.

Step-by-step solution

- Write down the equations

The given equations are: 1. \( \frac{a}{2} + \frac{b}{3} = 3 \) 2. \( a - b = 3 \)

- Clear denominators in the first equation

To eliminate the fractions, multiply the first equation by 6: \[ 6 \times \left( \frac{a}{2} + \frac{b}{3} \right) = 6 \times 3 \] This simplifies to: \[ 3a + 2b = 18 \]

- Use the second equation

The second equation is: \[ a - b = 3 \]

- Solve the system of linear equations

We now have two new equations: 1. \ 3a + 2b = 18 \ 2. \ a - b = 3 \ First, solve the second equation for \(a\): \[ a = b + 3 \] Next, substitute \(a = b + 3\) into the first equation: \[ 3(b + 3) + 2b = 18 \] Simplify and solve for \(b\): \[ 3b + 9 + 2b = 18 \] \[ 5b + 9 = 18 \] \[ 5b = 9 \] \[ b = 1.8 \]

- Find the value of \(a\)

Substitute \(b = 1.8\) back into \(a = b + 3\): \[ a = 1.8 + 3 = 4.8 \]

- Calculate \(a + b\)

Now add \(a\) and \(b\) to find \(a + b\): \[ a + b = 4.8 + 1.8 = 6.6 \]

Key concepts

GMAT linear equations

Linear equations are essential in GMAT math problems. They help to form relationships between variables. A linear equation is any equation that can be written in the form: ax + by = c where,

• a, b, and c are constants
• x and y are variables

For example, one of the equations in this problem is written as: a - b = 3 which is a typical linear equation. Linear equations are used to model real-world phenomena or solve mathematical problems where relationships between quantities need to be determined.

System of linear equations

When working with GMAT linear equations, you will often need to solve a system of linear equations, which consists of two or more linear equations involving the same set of variables. For example, the given exercise includes the following system of linear equations: \( \frac{a}{2} + \frac{b}{3} = 3 \) and \( a - b = 3 \). To solve this, you need to find the values of the variables that satisfy both equations simultaneously.
One common method is to use substitution or elimination. In this solution, we first get rid of fractions by multiplying through by a common denominator and then use substitution to solve for the variables.

GMAT algebra

Algebra is a crucial segment of the GMAT quantitative section. It involves manipulating equations and expressions to find unknown values. In this exercise, solving the linear equations involves several algebraic steps:

• Clearing the fractions by multiplying the entire equation by a common factor.
• Using substitution: solving one equation for one variable and substituting that expression into the other equation to find the values of both variables.
• Simplifying expressions: combining like terms and solving for the variable.

Working through each step methodically ensures that you don’t make arithmetic errors and can find the correct solution.

GMAT quantitative section

The GMAT quantitative section tests your mathematical reasoning, problem-solving ability, and knowledge of fundamental math concepts. Important areas covered include arithmetic, algebra, and geometry. This section includes both problem-solving and data sufficiency questions. In problem-solving questions like the one in this exercise, you'll be asked to apply mathematical principles to solve a specific problem. Key skills assessed include manipulating equations, understanding and working with fractions, and expressing mathematical relationships effectively.
To excel, practice regularly and focus on understanding the underlying mathematical concepts. Utilize practice exercises to become familiar with different question types and improve your speed and accuracy.