Problem 42 - Algebra 2

Question

Mr. Talbot is writing a science test. It will have true/false questions worth 2 points each and multiple-choice questions worth 4 points each for a total of 100 points. He wants to have twice as many multiple-choice questions as true/false. If most of his students can answer true/false questions within 1 minute and multiple-choice questions within 1\(\frac{1}{2}\) minutes, will they have enough time to finish the test in 45 minutes?

Step-by-Step Solution

Answer

Students have enough time; they need 40 minutes and have 45 minutes available.

Step 1: Define Variables

Let \( x \) represent the number of true/false questions and \( y \) represent the number of multiple-choice questions. We know that each true/false question is worth 2 points and each multiple-choice question is worth 4 points.

Step 2: Set Up Equations

From the problem, we have two equations:1. Based on the points: \( 2x + 4y = 100 \)2. Based on the number of questions: \( y = 2x \)

Step 3: Substitute Equation

Substitute the second equation \( y = 2x \) into the first equation to eliminate \( y \).\[ 2x + 4(2x) = 100 \]

Step 4: Simplify and Solve for \( x \)

Substitute and simplify the equation:\[ 2x + 8x = 100 \]\[ 10x = 100 \]\[ x = 10 \]This means there are 10 true/false questions.

Step 5: Solve for \( y \)

Using the relationship \( y = 2x \):\[ y = 2(10) \]\[ y = 20 \]There are 20 multiple-choice questions.

Step 6: Calculate Time Needed

Determine how much time is needed:* True/false: \( 10 \times 1 \text{ minute} = 10 \text{ minutes} \)* Multiple-choice: \( 20 \times 1.5 \text{ minutes} = 30 \text{ minutes} \)* Total time: \( 10 + 30 = 40 \text{ minutes} \)

Step 7: Determine if Time is Sufficient

Compare the total time required for the test (40 minutes) with the available time (45 minutes). Since 40 minutes required is less than 45 minutes available, students will have enough time to finish the test.

Key Concepts

Systems of Equations

Sometimes, two equations linked together hold the solution to a real-life problem. We call this linkage "systems of equations." They come into play when dealing with more than one variable. In the exercise above, Mr. Talbot wants to create a balanced test with true/false and multiple-choice questions. This scenario naturally leads to a system of equations:

• One equation based on the points per question: \(2x + 4y = 100\).
• Another equation showing the relationship between the number of questions: \(y = 2x\).

By solving these equations together, you can find how many of each question type Mr. Talbot should include. Employing systems of equations helps break down complex questions into manageable steps, solving multiple problems at once.

Substitution Method

When tackling systems of equations, the substitution method is a mighty tool. It involves replacing one variable with another to simplify the solution. In Mr. Talbot's challenge, we use substitution to find how many questions there are of each type. Here's how it works:

• First, take an easy-to-solve equation: \(y = 2x\).
• Then, substitute \(2x\) for \(y\) in the more complex equation: \(2x + 4(2x) = 100\).

After substitution, you get an equation with just one variable, \(x\), making it much easier to solve. This single-variable equation will guide you to the solution step by step. Finally, plug the solution back into one of the original equations to find the other variable. Using substitution keeps everything clear and organized, preventing errors.

Time Management in Exams

Being mindful of time in exams is crucial, especially when different types of questions are involved. True/false questions are typically quicker, needing only a minute each. Meanwhile, multiple-choice questions usually take longer as they need more thinking, around 1.5 minutes each.

Calculating total time for Mr. Talbot's exam means checking if students can finish within 45 minutes. Watch the math:

• Time for true/false: \(10 \times 1 = 10\) minutes.
• Time for multiple-choice: \(20 \times 1.5 = 30\) minutes.
• Total time for the test: 10 + 30 = 40 minutes.

Since 40 minutes is less than 45 minutes, students will have enough time. Managing your time well can ensure you don't miss any questions and can maximize your test performance, giving you that important boost in exams. Always remember to allocate your time wisely, leaving room to review your answers if needed.