Question 982656
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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?
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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 = {{{(99-7x)/2}}}.      (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 = {{{(99-7*13)/2}}} = 3


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


<U>ANSWER</U>.  13 correct, 3 wrong and 4 omitted.
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