Problem 19 - Intermediate Algebra

Question

Spring Tours. \(\quad\) A group of junior high students will be touring Washington, D.C. Their chaperons will have the \(\$ 1,810\) cost of the tour reduced by \(\$ 15.50\) for each student they personally supervise. How many students will a chaperon have to supervise so that his or her cost to take the tour will be \(\$ 1,500 ?\)

Step-by-Step Solution

Answer

A chaperon must supervise 20 students.

Step 1: Understand the Problem

The problem states that the tour cost of a chaperon reduces by $15.50 for each student they supervise. We need to determine how many students a chaperon must supervise for the tour cost to become $1,500. Initially, the tour cost is $1,810.

Step 2: Set Up the Equation

If a chaperon supervises \(x\) students, the cost reduction is \(15.50 times x\). The original cost is \(1,810, and we want the cost to decrease to \)1,500. Form the equation from this understanding: \[1810 - 15.5x = 1500\]

Step 3: Solve the Equation for x

First, subtract \(1,500 from both sides of the equation: \[1810 - 1500 = 15.5x\]Calculate the left side: \[310 = 15.5x\]Next, solve for \)x\( by dividing both sides by \)15.5$: \[x = \frac{310}{15.5}\]

Step 4: Calculate the Result

Perform the division:\[x = 20\]This means a chaperon must supervise 20 students for the cost to be $1,500.

Key Concepts

Problem Solving

Problem solving is the art of finding an effective solution to any kind of challenge, like the one faced by the chaperons in our problem. Here, we have a practical situation: reducing the cost of a tour based on the number of students supervised by each chaperon. To tackle such problems, it's important to first fully understand what's being asked and identify the known elements. We know the initial cost is $1,810, and we want it to reduce to $1,500 by supervising a certain number of students. This requires setting up a systematic approach:

• Understand the goal: Reduce the tour cost to $1,500.
• Identify influencing factors: The number of students supervised.
• Translate this understanding into mathematical concepts.

By logically breaking down what's given and what's needed, problem solving becomes more manageable. With each step, clarity paves the way towards finding an effective solution. With these specifics in mind, we're ready to move towards forming an equation that will guide us to the answer.

Equation Formation

Equation formation is all about translating real-world scenarios into mathematical expressions. For our problem, we need to convert the situation involving reduced costs and supervised students into an equation. This involves identifying the relevant variables and constants:

• The Total Cost: Initially \(1,810.
• Reduction per Student: \)15.50 per student.
• Desired Cost: \(1,500.

To form an equation, we note that each student supervised reduces the cost by \)15.50. If a chaperon supervises \(x\) students, the total reduction is \(15.5x\). Subtracting this reduction from the initial cost gives us a simple equation:\[1810 - 15.5x = 1500\]This equation captures the relationship between the elements of the problem and sets the stage to solve for \(x\), the number of students needed to achieve the target cost reduction.

Step-by-Step Solution

The step-by-step solution is where we solve the equation formed by methodically examining each step. Let's sequence the steps required to find out just how many students a chaperon needs to supervise:

• Step 1: Begin with the equation: \(1810 - 15.5x = 1500\).
• Step 2: Isolate the term with \(x\) by subtracting \(1,500 from \)1,810: \(1810 - 1500 = 310\).
• Step 3: You're left with: \(15.5x = 310\).
• Step 4: Solve for \(x\) by dividing both sides by \(15.5\): \(x = \frac{310}{15.5}\).
• Step 5: Calculate the result: \(x = 20\).

With each logical step, the problem reveals its solution. Our methodical approach shows that a chaperon would need to supervise 20 students for the trip cost to reduce to $1,500. With practice, following steps like these can make any equation-solving problem feel straightforward.