Question

The raw score on a certain standardized test is determined by subtracting \(\frac{1}{4}\) of the number of incorrect answers from the number of correct answers. If a student answered 30 questions and received a raw score of 20, how many questions did the student answer incorrectly?

Short answer

The student answered 8 questions incorrectly.

Step-by-step solution

Define the variables

Let x represent the number of correct answers, and y represent the number of incorrect answers. We know the total number of questions is 30, so: \(x + y = 30\) We also know that the raw score (20) is determined by subtracting \(\frac{1}{4}\) of the incorrect answers (y) from the correct answers (x): \(x - \frac{1}{4}y = 20\)

Solve the system of equations

We have a system of two linear equations: 1. \(x + y = 30\) 2. \(x - \frac{1}{4}y = 20\) We can solve this system by using the substitution method or the elimination method. In this case, we'll use the substitution method. From equation 1, we can express x in terms of y: \(x = 30 - y\) Now, substitute this expression for x in equation 2: \((30 - y) - \frac{1}{4}y = 20\)

Solve for y

Now, we will solve the equation for y: \((30 - y) - \frac{1}{4}y = 20\) Combine the y terms and subtract 30 from both sides: \(-y - \frac{1}{4}y = -10\) Combine the coefficients of y: \(-\frac{5}{4}y = -10\) Divide both sides by -\(\frac{5}{4}\) to isolate y: \(y = \frac{-10}{-\frac{5}{4}}\) Simplify the fraction on the right-hand side, and we find the number of incorrect answers: \(y = 8\)

Conclusion

The student answered 8 questions incorrectly.

Key concepts

Raw Score Determination

Understanding how a raw score is determined on standardized tests is crucial for test-takers. The raw score typically represents the number of correct responses adjusted by a penalty for incorrect answers. In many tests, such as the SAT, raw scores are computed by subtracting a fraction of the number of incorrect answers from the total number of correct answers. This penalizing factor exists to prevent random guessing.

For instance, the exercise demonstrates a common type of raw score calculation. If a student gets a raw score of 20 after answering 30 questions, meaning some of the answers were incorrect, to find out how many were wrong, we utilize a simple formula that accounts for this penalty. In this case, for every incorrect answer, \( \frac{1}{4} \) of a point is subtracted from the total score. The importance of understanding raw score determination is evident—it allows students to plan better test-taking strategies, like whether or not to guess on questions they're unsure about.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra that lays the groundwork for understanding more complex mathematical concepts. Linear equations describe a straight line when plotted on a graph and usually appear in the form of \( ax + b = c \) where \( a \) and \( b \) are constants, and \( x \) is the variable.

In the textbook exercise, we approach a system of linear equations represented by two different formulas. To solve such a system, there are several methods available, such as substitution and elimination. The substitution method involves expressing one variable in terms of the other, and then replacing that expression in the second equation to solve for the remaining variable.

Subsequently, ensuring a grasp of these equations is not just essential for test problems but also equips students with problem-solving skills applicable to real-world situations like finance and engineering.

SAT Math Problems

SAT math problems are designed to assess a student's proficiency in mathematical concepts typically taught in high school. These problems often involve algebra, geometry, and basic trigonometry, among other areas. The SAT Math section contains a variety of question types, including multiple-choice and grid-in responses.

Students preparing for the SAT should be comfortable with solving linear equations as these appear frequently in the test. Additionally, understanding how raw scores are calculated can help students strategize their pacing and guessing approach. It's advisable for students to attempt practice problems that mimic the difficulty and style of actual SAT questions, as this enhances their problem-solving speed and accuracy under timed conditions.

Incorporating realistic SAT math problems in preparation, like the one in our exercise where the student determines incorrect answers based on their raw score, enables the learners to familiarize themselves with the exam's format, improving their confidence and performance.