Question

A teacher grades students' tests by subtracting twice the number of incorrect responses from the number of correct responses. If Student A answers each of the 100 questions on her test and receives a score of $73,$ how many questions did Student A answer correctly? A 55 B 60 C 73 D 82 E 91

Short answer

91

Step-by-step solution

Identify Variables

Let’s define the variables. Let $x$ be the number of correct responses and $y$ be the number of incorrect responses.

Set Up Equations

Student A answers 100 questions in total. Therefore, $x+y=100$. The scoring formula is given by: $x-2y=73$.

Solve for One Variable

Rearrange the first equation to express $y$ in terms of $x$: $y=100-x$.

Substitute and Solve

Substitute $y=100-x$ into the second equation: $$x-2(100-x)=73$$. This simplifies to: $$x-200+2x=73$$ $$3x-200=73$$.

Solve for x

Isolate $x$ by adding 200 to both sides: $$3x=273$$. Divide by 3: $$x=91$$.

Verification

To verify, compute $y$: $$y=100-91=9$$. Check the score calculation: $$91-2(9)=73$$, which is correct.

Key concepts

Linear Equations

Linear equations are mathematical statements that show the equality of two expressions. They involve variables raised only to the first power. The exercise uses two linear equations to find the number of correct and incorrect answers.

We define variables for correct and incorrect answers: let $x$ be the number of correct answers and $y$ be the number of incorrect answers. The sum of both correct and incorrect answers equals the total number of questions:
$$x+y=100$$
The scoring formula subtracts twice the incorrect answers from the correct answers:
$$x-2y=73$$
By solving these equations together, we can find the exact values of $x$ and $y$.

Test Scoring Methods

This problem showcases a common test scoring method where instructors balance correct answers against incorrect ones to avoid guessing. Here, each correct answer equals one point, and each incorrect answer subtracts two points:

• Correct Answer: +1 point
• Incorrect Answer: -2 points

The final score is derived by the equation:
$$\text{Score}=\text{Correct Answers}-2\times \text{Incorrect Answers}$$
Student A's final score was given as $73$, which was based on their correct and incorrect answers according to this method.

Problem-Solving Steps

When tackling complex math problems, it helps to break them down into smaller steps. Let's see this in action for finding the number of correct answers student A got:

• 1. Identify Variables: Define $x$ as correct answers and $y$ as incorrect answers.
• 2. Set up Equations: Use the given information to form equations: $x+y=100$ and $x-2y=73$.
• 3. Solve for One Variable: Rearrange $x+y=100$ to get $y=100-x$.
• 4. Substitute and Solve: Replace $y$ in the second equation: $x-2(100-x)=73$.
• 5. Simplify: Solve $3x-200=73$ and isolate $x$.
• 6. Verification: Verify by computing $y$, ensuring results fit all given conditions. We found $x=91$. Verify correctness by checking if the calculated score matches the given score of $73$.

This systematic approach eliminates confusion and guides you to the solution clearly.

Algebra

Algebra is the branch of mathematics that deals with variables and the rules for manipulating these variables in formulas and equations. In this exercise, algebraic methods help us find the correct and incorrect answers.

Using algebraic substitution and elimination, we solve equations. For instance, we substituted $y=100-x$ into $$x-2y=73$$ to form a single equation in terms of $x$:

$$x-2(100-x)=73$$

Simplifying this, we got:
$$3x-200=73$$ and then solved for $x$:
$$3x=273$$

Finally, dividing by 3, we find:
$$x=91$$
Thus, algebra guided us to the solution efficiently, showcasing its power in problem-solving across various mathematical contexts.