# Word Problem help

#### dannyc

Each contestant at a game show starts with 30 points. A contestant earns 10 points for each question answered correctly and loses 5 points for each question answered incorrectly. Mitch answered twice as many questions correctly as incorrectly, finishing with 150 points. How many question did he answer correctly?

Why is it 16???

#### dokrbb

what did you try so far, it seems to be a two variable equation

#### HallsofIvy

MHF Helper
Let x be the number of questions he answered correctly, y the number of questions he answered incorrectly. Then his score is 10x- 5y= 150. An obvious first step is to divide through by 5 to get 2x- y= 30. As dokrbb says, that is a single equation with two variables. In general we could choose any number for x and calculate the corresponding y but because these are "numbers of questions", x and y must be non-negative integers so this is a "Diophantine" equation and we can find a formula for the possible solutions.

+ First note that if x= y= 30, we have 2(30)- 30= 30 so x= 30, y= 30 is one answer. But we can see that x= 30+ n, y= 30+ 2n is also a solution for any n: 2(30+ n)- (30+ 2n)= 60+ 2n- 30- 2n= 30. Since y cannot be negative, n cannot be smaller than -15. So we have an infinite number of solutions, x= 30+ n, y= 30+ 2n for $$\displaystyle n/ge -15$$ or, if you prefer, x= 15+ n, y= 2n for $$\displaystyle n\ge 0$$.

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#### emakarov

MHF Hall of Honor
Mitch answered twice as many questions correctly as incorrectly
Therefore we have

2x - y = 30
x = 2y

But the solution for x is not 16.

#### Plato

MHF Helper
Each contestant at a game show starts with 30 points. A contestant earns 10 points for each question answered correctly and loses 5 points for each question answered incorrectly. Mitch answered twice as many questions correctly as incorrectly, finishing with 150 points. How many question did he answer correctly?

Why is it 16???
Both replies #3 & #4 missed that bit of the question.

$$\displaystyle \\10x-5y+30=150\\x=2y$$

• 3 people