Question

A company paid \(\$ 500,000\) in merit raises to employees whose performances were rated \(A, B,\) or \(C .\) Each employee rated \(A\) received twice the amount of the raise that was paid to each employee rated \(C ;\) and each employee rated \(B\) received one-and-a-half times the amount of the raise that was paid to each employee rated \(C\), If 50 workers were rated \(A, 100\) were rated \(B,\) and 150 were rated \(C,\) how much was the raise paid to each employee rated \(A\) ? \(\$a. 370\) \(\$b. 625\) \(\$c. 740\) \(\$d. 1,250\) \(\$e. 2,500\)

Short answer

\$2,500

Step-by-step solution

Setting up the equation

Considering that the number of employees rated A is 50, B is 100 and C is 150 and the raise for each is really \(2x, 1.5x, x\) respectively. These all add up to the total raise, 500,000. The equation would thus be \(50(2x) + 100(1.5x) + 150x = 500,000\).

Simplifying the equation

By performing the multiplication and combining like terms, the equation becomes \(100x + 150x + 150x = 500,000\), which simplifies to \(400x = 500,000\).

Solving for \(x\)

Having established the simplified equation, we now divide both sides by 400 to isolate \(x\). This gives \(x = 500,000/400 = 1,250\)

Solving for the raise rated \(A\)

After finding the value of \(x\), we substitute it into the equation for the raise of A rating, which is \(2x\), resulting in \(2(1,250) = 2,500\).\(

Key concepts

Algebra

Algebra is at the core of solving many quantitative problems, especially when it involves setting up and simplifying equations to find unknown values. In this exercise, we start by defining variables to represent unknowns, then establish relationships between these variables using an equation.

Here, the merit raise amounts given to the employees rated A, B, and C were translated into algebraic expressions: each A rating got twice the C rating, and each B rating got 1.5 times. Therefore, using an unknown variable, say \(x\), to represent the amount for each employee rated C, the equations were:

- A's raise: \(2x\)
- B's raise: \(1.5x\)
- C's raise: \(x\)

Knowing the total number of employees and their corresponding ratings helped set up this equation. Algebraic expression allowed us to connect these variables and ultimately solve for the amount each category of workers received. Formulating and solving such equations is a fundamental algebra skill that simplifies the complexity of distribution problems.

Quantitative Reasoning

Quantitative reasoning involves understanding numerical data and relationships. In this exercise, it helps to interpret and utilize information given about employees’ ratings and related raises. This involves several steps: recognizing quantities, performing calculations, and making logical deductions.

The problem provided specific relationships between the raises based on performance ratings. Quantitative reasoning requires seeing the ratio relationships and translating them into mathematical terms. Here:

• Each A-rated employee gets a raise of \(2x\).
• Each B-rated employee gets a raise of \(1.5x\).
• Each C-rated employee gets a raise of \(x\).

With a clear concept of ratios and the total amount distributed, you rely on quantitative reasoning to solve: constructing an equation that represents the distribution of the \(\$500,000\) in raises. By breaking down and rearranging these relationships, you can derive the unknown value \(x\), representing the raise for C-rated employees.

Problem Solving

Problem solving is about sequentially breaking down a complex task into manageable parts to reach a solution. In this exercise, the problem is tackled step by step to simplify calculations and find results accurately.

We begin by analyzing the problem's context and identifying what information is given and what is needed. Next, a mathematical model representing the given situation is constructed through an equation. Problem-solving techniques include:

• Identifying and defining variables: Here, \(x\) to represent the smallest raise.
• Formulating the equation based on variables and given numbers.
• Simplifying and solving the equation to find the variable \(x\).
• Calculating specific quantities, like the raise for A-rated employees, using new data.

Once \(x\) is found, substituting it back yields the solution. This problem-solving process reflects analytical thinking, logical progression, and mathematical accuracy, essential for tackling GRE math problems efficiently.