Problem 55 - Prealgebra

Question

Several students on campus decide to start a small business that offers tutoring services to students enrolled in mathematics courses. The following equation shows the amount of money they collected at the end of the month, assuming expenses of \(\$ 400\) and a charge of \(\$ 30\) an hour per student. Solve the equation \(\$ 500=\$ 30 x-400\) to determine the number of hours students came in for tutoring in one month.

Step-by-Step Solution

Answer

The students tutored for 30 hours in one month.

Step 1: Understand the Equation

We have the equation that represents the total money collected: \(500 = 30x - 400\). Here, 500 is the net money collected, 30 is the rate charged per hour, \(x\) is the number of hours, and 400 is the expenses.

Step 2: Isolate the Variable Term

To solve for \(x\), we first need to isolate the term with \(x\). We do this by adding 400 to both sides of the equation:\[ 500 + 400 = 30x - 400 + 400 \]which simplifies to:\[ 900 = 30x \].

Step 3: Solve for x

Now, we need to solve for \(x\) by dividing both sides of the equation by 30:\[ \frac{900}{30} = \frac{30x}{30} \]which simplifies to:\[ x = 30 \].

Key Concepts

Exploring Mathematics Tutoring

Mathematics tutoring is an excellent resource for students who require additional support and guidance to enhance their understanding of mathematical concepts. In many schools and campuses, tutoring services are provided by peer groups or professionals, creating a conducive learning environment. With a one-on-one approach, tutors can cater to the specific needs of the students, making complex equations seem much simpler and less intimidating.
In this particular exercise, students decided to offer math tutoring services on their campus. Such endeavors not only help students grasp difficult concepts but also allow tutors to reinforce their knowledge and gain teaching experience. Moreover, it presents a real-life application of the very mathematical principles being applied, as students think critically about how to set fair rates and manage finances within their tutoring business.
When looking at how much to charge for tutoring, understanding common pricing models and what affects cost per hour can be beneficial. Pricing should cover any expenses while also being competitive enough to attract students. If students receive quality tutoring at a fair price, both the tutors and the students can thrive.

Diving into Algebra Concepts

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols to solve equations. It forms the foundation for more advanced mathematics and is crucial for developing problem-solving skills.
In this exercise, the main algebra concept is to solve a linear equation of the form \( ax + b = c \). This involves understanding each component of the equation:

• \( a \) is the coefficient of the variable \( x \), representing the rate per hour, which is \( 30 \) in this case.
• \( x \) is the variable that needs to be solved. It denotes the number of hours students attended tutoring.
• \( b \) represents any additional constants or expenses, set at \( 400 \).
• \( c \) is the net amount collected, \( 500 \) here.

Identifying these elements allows us to reframe the equation as a problem of isolating \( x \) and discovering its value through logical steps.
Grasping these algebraic concepts is essential as it enables students to apply mathematical theory to real-world scenarios, enhancing both their educational and practical understanding.

Mastering the Equation Solving Process

Solving an equation involves a systematic approach to finding an unknown variable that satisfies the equation's condition. This requires various steps, as demonstrated in the step-by-step solution provided.
The process begins with isolating the term that contains the unknown variable. In our example, we add \( 400 \) to both sides, transforming the equation into \( 900 = 30x \). This step is crucial because it simplifies the equation and brings us closer to the solution.
After that, the next step is to solve for \( x \) by performing the operation that will make \( x \) the subject of the equation. Here, dividing both sides by \( 30 \) provides us with \( x = 30 \).
It’s important to remember a few tips when solving equations:

• Perform the same operation on both sides to maintain the equation's balance.
• Carefully follow arithmetic rules to avoid errors.
• Check your solution by substituting it back into the original equation to ensure it satisfies all parts.

Consistency in the equation solving process builds confidence and skill, making tackling even complex problems achievable. Understanding each step encourages students to handle algebraic tasks effectively in both academic and real-world settings.