https://file.aiquickdraw.com/mathbot/file/1725507809156jk044mhl.jpg

The problem provided involves Carol, who has 50 minutes to take an English test with two types of questions: multiple-choice and short-answer. There are 20 multiple-choice questions and 20 short-answer questions. The details include:

- Each multiple-choice question takes \(1\frac{1}{2}\) minutes (which is 1.5 minutes).
- Each short-answer question takes 2 minutes.
- Each correct multiple-choice question earns 2 points.
- Each correct short-answer question earns 3 points.
- Any question Carol answers, she gets correct.

The problem asks:

**(a)** What is the maximum possible score she can receive?  
**(b)** What advice might you give Carol to improve her score?

Let's first determine the maximum score Carol can achieve and then advise how she can optimize her score based on the time constraints.

### Step 1: Calculate Time for All Questions

- Time for each multiple-choice question: \( 1.5 \) minutes.
  \[
  \text{Time for all multiple-choice questions} = 20 \times 1.5 = 30 \, \text{minutes}.
  \]

- Time for each short-answer question: \( 2 \) minutes.
  \[
  \text{Time for all short-answer questions} = 20 \times 2 = 40 \, \text{minutes}.
  \]

- Total time required to answer all 40 questions:
  \[
  30 \, \text{minutes} + 40 \, \text{minutes} = 70 \, \text{minutes}.
  \]

Carol only has 50 minutes, so she cannot answer all the questions. She needs to prioritize to maximize her score.

### Step 2: Maximize the Score

Each multiple-choice question gives 2 points, and each short-answer question gives 3 points. To maximize her score, Carol should focus on short-answer questions first, as they offer more points per question.

- Time available: 50 minutes.
- Time to complete one short-answer question: 2 minutes.
  \[
  \text{Maximum number of short-answer questions she can complete} = \frac{50}{2} = 25 \, \text{questions}.
  \]
  But there are only 20 short-answer questions.

### Step 3: Calculate the Maximum Score

- If Carol answers all 20 short-answer questions, she will spend:
  \[
  20 \times 2 = 40 \, \text{minutes}.
  \]
  Her score from these questions will be:
  \[
  20 \times 3 = 60 \, \text{points}.
  \]
  
- After answering all short-answer questions, she has:
  \[
  50 \, \text{minutes} - 40 \, \text{minutes} = 10 \, \text{minutes} \text{ left}.
  \]
  In 10 minutes, she can answer:
  \[
  \frac{10}{1.5} \approx 6 \, \text{multiple-choice questions}.
  \]
  Her score from these multiple-choice questions will be:
  \[
  6 \times 2 = 12 \, \text{points}.
  \]

- **Total maximum score**:
  \[
  60 + 12 = 72 \, \text{points}.
  \]

### Step 4: Advice to Improve Her Score

To maximize her score, Carol should:
1. Prioritize answering all short-answer questions (since they provide more points per minute).
2. Use any remaining time to answer as many multiple-choice questions as possible.

### Summary:
**(a)** The maximum possible score Carol can receive is 72 points.  
**(b)** The advice is to first focus on answering all the short-answer questions, then use the remaining time to answer multiple-choice questions.

Would you like further details on any specific part? Here are some follow-up questions for deeper understanding:

1. How would the strategy change if the time for answering short-answer questions were different?
2. What if Carol were better at multiple-choice questions and got them all correct faster?
3. Could Carol finish more questions if the test was organized differently?
4. How can we calculate the most efficient use of her time?
5. What would happen if she had more total time?

**Tip:** Focus on maximizing points per minute when dealing with time-limited exams.

Carol Sommers has 50 minutes to take an English test that has 20 multiple-choice questions and 20 short-answer questions. She knows she can answer a multiple-choice question in 1.5 minutes and a short-answer question in 2 minutes. Each correct multiple-choice answer receives 2 points, and each correct short-answer answer receives 3 points. What is the maximum possible score she can receive? What advice might you give Carol to improve her score?

This problem explores maximizing a test score under time constraints with 20 multiple-choice and 20 short-answer questions. Each question type has a different time requirement and point value. Learn the step-by-step approach to find the best strategy for Carol Sommers.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation