Problem 55 - Algebra 1: Concepts and Skills

Question

Your teacher is giving a test worth 250 points. There are 68 questions. Some questions are worth 5 points and the rest are worth 2 points. How many of each question are on the test? (Lesson 7.4)

Step-by-Step Solution

Answer

There are 38 questions that are worth 5 points and 30 questions that are worth 2 points

Step 1: Define Variables

Let's use variables to represent the unknowns in our problem. Let \(x\) represent the number of questions worth 5 points and let \(y\) represent the number of questions worth 2 points.

Step 2: Set up the Equations

We can set up two equations from the problem statement.Equation 1 corresponds to the total number of questions, i.e., \(x + y = 68\)Equation 2 corresponds to the total points, i.e., \(5x + 2y = 250\)

Step 3: Solve for one of the Variables

Solving equation 1 for \(x\), we get \(x = 68 - y\).

Step 4: Substitute

Replace \(x\) from equation 3 into equation 2. This gives \(5(68 - y) + 2y = 250\). Simplifying the equation, we get \(340 - 5y + 2y = 250\), further simplifying, we get \(3y = 90\).

Step 5: Solve for \(y\)

Solving for \(y\) from equation 4 gives \(y = 30\).

Step 6: Solve for \(x\)

Substitute \(y = 30\) into equation 1, we get \(x = 68 - 30 = 38\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and are used to solve mathematical problems involving unknown quantities. In our exercise, these expressions are seen when we define variables to represent unknown values. Here, we use expressions like \(5x + 2y = 250\) to represent the relationship between the number of questions worth 5 points and 2 points, and their respective total points.

With algebra, we convert word problems into mathematical expressions, making it easier to find solutions. It's like translating a language of numbers and operations into symbols. Each algebraic expression represents an equation that helps describe a real-life scenario mathematically.

When dealing with algebraic expressions, remember:

• Identify the unknowns and use variables to represent them.
• Set up expressions based on the information given.
• Ensure each expression correctly represents the arrangement of values.

This systematic approach to formulating expressions is essential for solving equations.

Solving Equations

Solving equations involves finding the values of the variables that make the equation true. In the exercise, we were tasked with solving simultaneous equations. The first step is usually to simplify the given equations or rearrange them to make one of the variables the subject.

Consider the equations from the exercise:

• \(x + y = 68\)
• \(5x + 2y = 250\)

These equations are solved by substitution or elimination methods. We chose to express \(x\) in terms of \(y\) from the first equation and substitute into the second equation. This simplifies the problem to one equation with one variable, making it easier to solve.

Simplification and substitution are powerful techniques:

• They help reduce complex problems to simpler ones.
• By focusing on one variable, solutions become clearer and more achievable.
• Verify your solutions by substituting back to ensure they satisfy the original equations.

Mathematical Problem-Solving

Mathematical problem-solving is a step-by-step process of identifying, analyzing, and finding solutions to problems. In the context of the exercise, our goal was to find out the number of questions worth different points.

Problem-solving starts with understanding the problem. You interpret the data provided and determine the relationships between different elements. With the test problem, identifying that there are two types of questions is crucial.

The process of defining variables, setting up equations, and systematically solving them exemplifies problem-solving in action. Here are steps often involved:

• Understand the problem: Thoroughly read and interpret the given information.
• Plan: Decide which strategies or operations to use (e.g., substitution or elimination).
• Execute the plan: Carry out calculations step-by-step.
• Review: Check your work to ensure all parts of the solution make sense.

Mastering these steps enhances your ability to solve a variety of mathematical problems efficiently.