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Math Help - system of equations word problem

  1. #1
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    system of equations word problem

    Mr. Day and Ms. Knight gave the same test consisting of 20 short-answer questions, but marked their test papers using different grading systems. Mr. Day awarded 5 points for each correct answer, made no deduction for an incorrect or omitted answer, and then added 10 points to the total. Ms. Knight gave 6 points for each correct answer and subtracted 1.5 points for each incorrect or omitted answer. Two students in different classes answered the same number of questions correctly and received the same test grade. Determine, using algabraic method, the test grade both students received. Confirm your answer graphically or by constructing a table using your graphing calculator.
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  2. #2
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    Quote Originally Posted by cityismine View Post
    Mr. Day and Ms. Knight gave the same test consisting of 20 short-answer questions, but marked their test papers using different grading systems. Mr. Day awarded 5 points for each correct answer, made no deduction for an incorrect or omitted answer, and then added 10 points to the total. Ms. Knight gave 6 points for each correct answer and subtracted 1.5 points for each incorrect or omitted answer. Two students in different classes answered the same number of questions correctly and received the same test grade. Determine, using algabraic method, the test grade both students received. Confirm your answer graphically or by constructing a table using your graphing calculator.
    We have three unknowns so we will need three equations to solve this system.

    let C be the number correct; I be the number incorrect; and S the score

    Mr Day: 5c+10=s
    Ms. Knight: 6c-1.5I=s
    question on quiz: c+I=20 \iff I=20-c


    setting the first to equal we get

    5c+10=6c-1.5I

    Now we can sub in I to get

    5c+10=6c-1.5(20-c) \iff 5c+10=6c-30+1.5c \iff 40=2.5c \iff 16=c

    We can now back sub in to find the other values.

    I=20-16=4 and s=5c+10=5(16)+10=90

    Yeah
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  3. #3
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    Hey, I was trying to solve this using two equations, but I guess you need three. Thanks for explaining it so thoroughly.
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  4. #4
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    Hello, cityismine!

    Mr. Day and Ms. Knight gave the same test consisting of 20 short-answer questions,
    . . but marked their test papers using different grading systems.
    Mr. Day gave 5 points for each correct answer, made no deduction for an incorrect
    . . or omitted answer, and then added 10 points to the total.
    Ms. Knight gave 6 points for each correct answer and subtracted 1.5 points for each
    . . incorrect or omitted answer.
    Two students in different classes answered the same number of questions correctly
    . . and received the same test grade.

    Determine, using algabraic method, the test grade both students received.
    Confirm your answer graphically or by constructing a table using your graphing calculator.

    The student in Mr. Day's class got x answers right.
    He got 5 points per correct answer plus 10 points.
    His grade was: . 5x + 10

    The student in Ms. Knight's class got x right and 20-x wrong.
    She got 6 points per correct answer: . 6x
    and lost 1.5 point for each wrong answer: . 1.5(20-x)
    Her grade was: . 6x - 1.5(20-x) \:=\:7.5x - 30

    The two grades were equal: . 5x + 10 \:=\:7.5x - 30 \quad\Rightarrow\quad 2.5x \:=\:40 \quad\Rightarrow\quad x \:=\:16

    Their grade was: . 5(16) + 10 \:=\:7.5(16) - 30 \:=\:\boxed{90}

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    Graph the lines: . \begin{array}{ccc}y &=& 5x+10 \\ y&=&7.5x - 30\end{array}

    They intersect at: (16,\,90)

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