SOLUTION: in a quiz game of 20 questions, each correct answer earns 5 points. Each incorrect answer reduces the score by 2 points. Omitted questions scored 0 points. Jana has a total score o

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Question 982656: in a quiz game of 20 questions, each correct answer earns 5 points. Each incorrect answer reduces the score by 2 points. Omitted questions scored 0 points. Jana has a total score of 59 points. How many questions did she omit?
Found 3 solutions by CubeyThePenguin, ikleyn, greenestamps:
Answer by CubeyThePenguin(3113)   (Show Source): You can put this solution on YOUR website!
59 points ==> 13 correct, 3 wrong, 4 omitted (65 - 6 + 0)


Answer by ikleyn(52946)   (Show Source): You can put this solution on YOUR website!
.
in a quiz game of 20 questions, each correct answer earns 5 points.
Each incorrect answer reduces the score by 2 points. Omitted questions scored 0 points.
Jana has a total score of 59 points. How many questions did she omit?
~~~~~~~~~~~~~~ 


Let x = #of correct;  y = # of incorrect;  z = # of omitted.

From the condition, we have these equations for integer numbers x, y, z

    x + y + z = 20           (1)

    5x - 2y   = 59           (2)

    x >= 0, y>= 0, z >= 0    (3)


From equation (2) and inequality y >= 0, we have  x >= 12.


Multiply equation (2) by 2 (both sides) and then add with equation (2).  You will get


    7x + 2z = 99,

          z = .      (4)


Now check, for which integer values of x from 12 to 13 inclusive you will have an integer value of z in formula (4).

The only such number of x is x = 13, giving z =  = 3


Then from equation (1)  y = 20 - 13 - 3 = 4.


ANSWER.  13 correct, 3 wrong and 4 omitted.



Answer by greenestamps(13219)   (Show Source): You can put this solution on YOUR website!


A quick informal solution using logical reasoning and some simple mental arithmetic -- if a formal algebraic solution is not required.

Each correct answer earns 5 points; the total points earned for the correct answers is a multiple of 5.

Each incorrect answer loses 2 points; the total points lost for the correct answers is an even number.

Jana's score was 59, an odd number. That means the number of correct answers has to be odd, making the total for the correct answers an odd multiple of 5 -- that is, a number with units digit 5.

So 59 has to be some even number less than a number with units digit 5.

The first possibility is 59 = 65-6, which means 13 correct answers an 3 wrong answers.

The next possibility is 59 = 75-16; but that requires far more than 20 questions on the test.

ANSWER: 13 right, 3 wrong; that means 4 questions were unanswered.
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