Question

Randy earns \(\$ 5\) for every order he fills at a call-in center, plus base pay of \(\$ 150\) per week. During a week in which he earns \(\$ 580\), how many orders does he fill? A. 86 B. 94 C. 116 D. 146

Short answer

Randy fills 86 orders.

Step-by-step solution

- Identify the variables

Let the number of orders Randy fills be denoted by the variable, say, 'x'.

- Set up the equation based on the problem

Randy's total earnings are composed of his base pay plus his earnings from filling orders. We can represent this relationship with the equation: \[ 150 + 5x = 580 \]

- Subtract base pay from total earnings

Subtract the base pay of \(150 from his total earnings of \)580 to find out how much he earned just from filling orders: \[ 580 - 150 = 430 \]

- Solve for the number of orders filled

To find the number of orders, divide the amount earned from filling orders by the amount he earns per order: \[ x = \frac{430}{5} = 86 \]

- Verify the solution

Ensure the solution is correct by checking: \[ 150 + 5 \times 86 = 150 + 430 = 580 \] This verifies that our solution is correct.

Key concepts

linear equations

A linear equation is an equation where the highest power of the variable is one. It forms a straight line when graphed on a coordinate plane. In this exercise, the linear equation represents the relationship between Randy's weekly earnings and the number of orders he fills. We've created a linear equation by setting Randy's base pay plus his earnings per order equal to the total amount he earned.

For example, the equation for this problem is: \[ 150 + 5x = 580 \]

Here, 'x' is the number of orders Randy filled. To solve a linear equation, you typically isolate the variable on one side, which involves basic algebraic operations like addition, subtraction, multiplication, or division.

problem solving

Problem-solving involves a logical sequence of steps to reach a solution. Let’s go through the steps used in this exercise to solve the problem:

• Identify the variable: We first identify 'x' as the number of orders Randy fills.
• Set up the equation: We translate the word problem into a mathematical equation.
• Isolate the variable: We perform operations to isolate 'x' and find its value.

Working through each step methodically helps break down complex problems into manageable parts. For instance, subtracting the base pay and then dividing by the earnings per order helps us isolate and solve for 'x'. Breaking the problem into smaller steps makes it easier to follow and understand.

algebra

Algebra is a branch of mathematics dealing with variables and the rules for manipulating these variables. In this problem, algebra helps us set up an equation based on Randy's earnings.

We start with the relationship between his total earnings and the number of orders he fills. Here's the algebraic breakdown: \ \ \

• We know Randy's base pay is \(150 and he earns an additional \)5 per order.
• The total amount he earned in a week is $580.

Using this information, we create the equation: \[ 150 + 5x = 580 \] The next steps involve standard algebraic techniques: \ \ \ * Subtract 150 from both sides: \[ 580 - 150 = 430 \] * Divide by 5 to solve for 'x': \[ x = \frac{430}{5} = 86 \]

Understanding algebra is crucial for forming and solving equations that represent real-world problems.