Each question sandra answers incorrectly changes her overall score by -(3)/(4) points. Sandra’s overall score was -4(1)/(2) points, and then she answered the last question incorrectly. What was sandra’s final score?
The Correct Answer and Explanation is:
To solve the problem, we need to determine how many questions Sandra answered incorrectly, given that each incorrect answer changes her score by –3/4 point and her overall score is –4 1/2 points.
Step 1: Convert the mixed number to an improper fraction
Sandra’s total score:
–4 1/2 = –(4 × 2 + 1)/2 = –9/2
Step 2: Let the number of incorrect answers be x
Each incorrect answer deducts 3/4 of a point. So if Sandra got x questions wrong:
Total score = –(3/4) × x
We are told that the total score is –9/2, so:
$$
-\frac{3}{4}x = -\frac{9}{2}
$$
Step 3: Solve for x
Multiply both sides of the equation by –1 to remove the negative signs:
$$
\frac{3}{4}x = \frac{9}{2}
$$
Now, multiply both sides by the reciprocal of 3/4 (which is 4/3):
$$
x = \frac{9}{2} \times \frac{4}{3} = \frac{36}{6} = 6
$$
✅ Final Answer: Sandra answered 6 questions incorrectly.
Explanation (300+ words):
This problem is a straightforward application of proportional reasoning using fractions. We’re told that every question Sandra gets wrong causes a deduction of 3/4 point, and that the final impact on her overall score was a negative 4 1/2 points. The goal is to figure out how many questions she must have gotten wrong to result in that total deduction.
To begin, we converted the mixed number –4 1/2 into an improper fraction to make calculations easier. Mixed numbers are often tricky in algebra, so switching them into improper fractions like –9/2 simplifies the equation setup.
We then wrote an algebraic expression that represents Sandra’s score deduction from incorrect answers. If she got x questions wrong, and each one deducts 3/4 of a point, then her total deduction is –(3/4)×x. Setting this equal to her actual total score of –9/2 allows us to solve for x.
At that point, the equation –(3/4)x = –9/2 is a simple linear equation. We eliminated the negatives and isolated x by multiplying by the reciprocal of 3/4, which is 4/3. This gives us:
$$
x = \frac{9}{2} \times \frac{4}{3} = 6
$$
This tells us that Sandra must have gotten 6 questions wrong to end up with a score of –4 1/2.
This problem is an excellent example of how understanding fraction multiplication, reciprocals, and converting between mixed and improper fractions can make solving real-world algebra problems easier.