Problem 70 - Algebra and Trigonometry

Question

Suppose that we agree to pay you \(8 \notin\) for every problem in this chapter that you solve correctly and fine you 5 e for every problem done incorrectly. If at the end of 26 problems we do not owe each other any money, how many problems did you solve correctly?

Step-by-Step Solution

Answer

The answer to how many problems were solved correctly is \(x\) problems, and the number of incorrectly solved problems is \(y\) problems (with \(x\) and \(y\) being the calculated values)

Step 1: Setup the System of Equations

First, setup the system of equations based on the information provided in the problem. Let \(x\) represent the correct problems and \(y\) represent the incorrect ones. From the problem, two information can be extracted: \n\n1) Total problems done are 26, which gives us the equation: \(x + y = 26\) \n\n2) The '8€' per correct and '5€' per wrong (fine) make a total of \(0€\), which gives us: \(8x - 5y = 0\)

Step 2: Solve the System of Equations

Next, solve for \(x\) and \(y\) systematically. By multiplying the first equation by 5, subtracting the second equation from the result, and zeroing out \(y\), you are left with an equation that contains only one variable, \(x\). This allows to find \(x\).

Step 3: Calculate the number of incorrect problems

Insert the value of \(x\) (correct problems) into the first equation: \(x + y = 26\) and solve it to get the value of \(y\) (incorrect problems). This calculation gives you the number of problems that you solved incorrectly.

Key Concepts

Algebraic Equations

Understanding algebraic equations is crucial for solving a wide range of mathematical problems. In our exercise, we began by identifying two algebraic equations. Algebraic equations are expressed as statements showing the equal value of two mathematical expressions using variables and constants. Typically, they use symbols such as \( x \) or \( y \) to represent unknowns.
In this particular problem:

• The first equation was based on the total number of problems: \( x + y = 26 \), where \( x \) is the number of correct problems, and \( y \) is the number of incorrect problems.
• The second equation was formed from the monetary values involved: \( 8x - 5y = 0 \). This equation reflects the condition where you are paid for solving a problem correctly and fined for incorrect answers.

These equations form a system which can be solved together to find our unknowns \( x \) and \( y \).

Problem Solving

Problem solving in mathematics often requires translating a word problem into a system of equations. These types of exercises help develop analytical thinking and the ability to abstract information. We started by translating the problem details into mathematical expressions.
Firstly, recognize the quantities involved and their relations. Here, it involved the total number of problems and financial transactions. Clearly understanding the problem is key.

• Next, you identify what you are solving for – in this case, the number of correct and incorrect problems.
• Finally, you apply logical reasoning to systematically solve the set equations.

This approach reveals insights into how algebra can be used for logical analysis and decision-making in solving real-world-like problems.

Correct and Incorrect Answers

When solving problems, differentiating between correct and incorrect answers often requires careful analysis. The problem asked us to find out how many problems were solved correctly.
For correct problems, you are rewarded monetarily, while incorrect problems result in a financial penalty. This creates a direct relation between the type of solution and the outcome (being paid or fined).
Realizing which answers contribute positively or negatively impacts the result, sharpening your decision-making skills. Ensuring correct answers requires understanding the problem's nuances and accurately setting up equations.

Setup Equations

Setting up equations is the fundamental first step in problem solving. It involves converting textual information into mathematical language.To set up the equations in our problem:

• First, identify what each variable will represent; here, \( x \) and \( y \) represent the number of correct and incorrect problems, respectively.
• Create equations representing each part of the problem. One equation tells us the sum of problems, and the other relates the monetary amounts linked to correct and incorrect answers.

When setting up equations, the accuracy is crucial. Done correctly, it leads to solving the problem smoothly. Ensuring clarity and breaking down the problem aids in visualizing and organizing information into solvable forms.