a quiz has 25 questions with 4points awarded for each correct answer and 1 point deducted for each incorrect answer,with zero for each question omitted.John scores 77points.how many questions did he omit?
number correct --- x
number incorrect --- y
number skipped ---- 25 - x - y
4x - y + 0(25-x-y) = 77
4x - y = 77
y = 4x - 77
so 4x - 77 ≥ 0 and we also know x+y ≤ 25
4x ≥ 77
x ≥ 77/4 , but x must be an integer
so x ≥20
so we could have
x y (s)kipped
20 3 2
21 7 -2 -----> invalid
22 11 -8 ----> invalide
etc
the only one that will work is
correct 20
wrong 3
skipped 2
To determine how many questions John omitted, we can use the information provided:
- Each correct answer is awarded 4 points.
- Each incorrect answer has 1 point deducted.
- Omitted questions do not contribute to the score (score is zero).
Let's assume John omitted "x" number of questions.
Now, let's calculate the total score:
Total Score = (Number of Correct Answers x 4) - (Number of Incorrect Answers x 1) + (Number of Omitted Questions x 0)
John's total score is given as 77 points.
77 = (Number of Correct Answers x 4) - (Number of Incorrect Answers x 1) + (x x 0)
77 = (Number of Correct Answers x 4) - (Number of Incorrect Answers x 1)
Since each question has only one possible outcome (correct, incorrect, or omitted), we can assume that the number of questions attempted is equal to the sum of the number of correct and incorrect answers.
Number of Questions Attempted = Number of Correct Answers + Number of Incorrect Answers
Therefore, we can rewrite the equation as:
77 = (Number of Questions Attempted x 4) - (Number of Questions Attempted - x x 1)
77 = (Number of Questions Attempted x 4) - (Number of Questions Attempted - x)
Simplifying further:
77 = 4 x Number of Questions Attempted - Number of Questions Attempted + x
Combining like terms:
77 = 3 x Number of Questions Attempted + x
We also know that:
Number of Questions = Number of Questions Attempted + Number of Omitted Questions
Substituting the value of Number of Questions in the equation:
77 = 3 x (Number of Questions - x) + x
77 = 3 x Number of Questions - 3x + x
77 = 3 x Number of Questions - 2x
Moving the terms:
3 x Number of Questions = 77 + 2x
3 x Number of Questions - 2x = 77
Since we don't know the exact number of questions, we cannot determine x directly from this equation. However, we can calculate the possible values of x based on the previous equation.
By trial and error, we can find that if x = 5, then the equation satisfies:
3 x Number of Questions - 2x = 77
3 x Number of Questions - 2(5) = 77
3 x Number of Questions - 10 = 77
3 x Number of Questions = 87
Number of Questions = 87 / 3
Number of Questions = 29
Since we assumed that John omitted "x" number of questions, and we found one possible solution where x = 5 and the total number of questions is 29, it means John omitted 5 questions.