To solve these problems, we'll use a system of equations. Let's denote:
C = number of questions answered correctly
I = number of questions answered incorrectly
B = number of questions left blank
(a) We know that Bernard's score is zero. From the information given, we can set up the equation:
6C - 4I = 0 (eq. 1)
We also know that there were a total of 20 questions on the quiz, so we have the equation:
C + I + B = 20 (eq. 2)
We want to find the value of C, so let's solve this system of equations.
From eq. 1, we can rearrange it as:
6C = 4I
Simplifying it further by dividing both sides by 2, we have:
3C = 2I (eq. 3)
Next, let's substitute eq. 3 into eq. 2:
C + I + B = 20
3C/2 + C + B = 20
(3C + 2C)/2 + B = 20
5C/2 + B = 20
Rearranging this equation, we get:
5C + 2B = 40 (eq. 4)
Now, we can solve equations 3 and 4 simultaneously to find the values of C and B.
Multiply eq. 3 by 5/3 to make the coefficients of C the same:
5C/3 = 10I/3
Substitute this value into eq. 4:
(5C/3) + 2B = 40
(10I/3) + 2B = 40
Multiplying through by 3 to remove the fractions, we have:
5C + 6B = 120 (eq. 5)
Now we can solve equations 3 and 5 simultaneously.
From eq. 5, rearrange it as:
5C = 120 - 6B
Substitute this into eq. 3:
120 - 6B = 2I
60 - 3B = I (eq.6)
Now we have two equations to solve simultaneously:
60 - 3B = I (eq. 6)
3C = 2I (eq. 3)
Let's look at some possible values for B and I:
If B = 0, then I = 60, but this is not a valid solution since there are only 20 questions in total.
If B = 1, then I = 57, but this is also not a valid solution.
If B = 2, then I = 54, still not a valid solution.
If B = 3, then I = 51, which means C = 34.
Therefore, Bernard answered 34 questions correctly.
(b) We know that Raymond scored 16 for the quiz, and the number of questions he answered correctly is the same as the questions he answered incorrectly. Let's call this number X.
So the equation is:
6X - 4X = 16
Simplifying this equation, we have:
2X = 16
Dividing both sides by 2, we find:
X = 8
Now we know that Raymond answered 8 questions correctly and 8 questions incorrectly. Let's call the number of questions left blank Y.
So the equation is:
8 + 8 + Y = 20
Simplifying this equation, we have:
16 + Y = 20
Subtracting 16 from both sides, we find:
Y = 4
Therefore, Raymond left 4 questions blank.