Problem 73 - Intermediate Algebra

Question

\(A\) student had a score of \(70 \%\) on a test that contained 30 questions. To improve his score, the instructor agreed to let him work 15 additional questions. How many of those must he get right to raise his grade to \(80 \% ?\)

Step-by-Step Solution

Answer

The student must get all 15 additional questions correct.

Step 1: Understand the Current Situation

The student had a score of 70%, which means he got 70% of the 30 questions correct. We can calculate the number of correct answers as \( 0.7 \times 30 = 21 \) questions.

Step 2: Calculate the Current Score

With 21 correct answers out of 30 questions, the student's initial score is 70%.

Step 3: Determine the Total Questions After Additional Questions

Adding the 15 additional questions brings the total number of questions to \( 30 + 15 = 45 \).

Step 4: Calculate the Required Correct Answers for 80%

To achieve a score of 80% on the 45 total questions, the student must answer \( 0.8 \times 45 = 36 \) questions correctly.

Step 5: Determine the Additional Correct Answers Needed

Since the student already has 21 correct answers, he needs \( 36 - 21 = 15 \) more correct answers among the additional 15 questions.

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. In this context, algebra helps us to calculate unknown values from known information. When a student wants to improve their test score from 70% to 80%, algebra is utilized to find out how many more questions they need to answer correctly. We start by setting up an equation based on the percentage calculations:

• Original correct answers: 21 out of 30 questions.
• New total questions: 30 (original) + 15 = 45 questions.

To find the number of questions needed to be correct for an 80% score, we use the equation: \[ 0.8 \times 45 = 36 \ \text{Correct answers needed after improvement: 36} \] This simple use of algebra allows us to solve practical problems like test score improvements.

Problem Solving

Problem solving is a critical aspect of learning, especially in mathematics. It involves identifying the issue, finding ways to address it, and applying solutions effectively. In our scenario, problem solving involves several clear steps:

• Identifying the current score as insufficient.
• Calculating the number of additional questions required.
• Determining how many additional correct answers are needed.

By patiently working through each step, the student gains clarity about the process. This systematic approach not only simplifies complex problems but also builds confidence in handling similar issues in the future.

Test Scores

Test scores are numerical representations of a student's performance on an exam. They are often expressed as percentages, which make it easier to understand and compare performance. In this case, the student's original score was 70%, derived from correctly answering 21 out of 30 questions.

• Original score calculation: \( 0.7 \times 30 = 21 \) correct answers.
• Goal score: 80% on 45 questions total.

Understanding test scores helps in setting realistic goals for improvement. The valuable insights gained from deconstructing this scenario can aid students in setting achievable targets for future assessments.

Proportion

Proportion involves creating a relationship between two ratios or fractions. It is used extensively in mathematics to solve problems regarding size or quantity comparisons. In the problem at hand, proportion is key to determining the relationship between the score gained from correct answers and the total questions.

• Initial proportion: 21 correct answers out of 30 questions.
• Desired proportion: 36 correct answers out of 45 questions (80%).

The use of proportion allows students to examine how changes in one part of a test (such as number of questions) affect the overall outcome. By understanding and applying proportions, lay learners can break down and solve percentage problems more easily.