There were 20 questions in a quiz. For each question answered correctly, 6 points were awarded. For each question answered incorrectly, 4 points were deducted. For each question left blank, no points were awarded or deducted.
(a) Bernard answered all the questions in the quiz but obtained a score of zero. How
many questions did Bernard answer correctly?
(b) Raymond scored 16 for the quiz. The number of questions he answered correctly is the same as the questions he answered incorrectly. How many questions did he leave blank?
Let's assume that Bernard answered x questions correctly.
He also answered (20 - x) questions incorrectly.
According to the given information, the total score he obtained is 0, so we can write the equation:
6x - 4(20 - x) = 0
6x - 80 + 4x = 0
10x - 80 = 0
10x = 80
x = 8
Therefore, Bernard answered 8 questions correctly.
Let's assume that Raymond answered x questions correctly.
According to the given information, he also answered x questions incorrectly.
The total score he obtained is 16, so we can write the equation:
6x - 4x = 16
2x = 16
x = 8
Therefore, Raymond answered 8 questions correctly.
Since the number of questions he answered correctly is the same as the questions he answered incorrectly, he answered a total of:
8 + 8 = 16 questions
There were 20 questions in total, so he left:
20 - 16 = 4 questions blank. Answer: \boxed{4}.
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.
To find the number of questions Bernard answered correctly and the number of questions Raymond left blank, we can set up equations based on the given information.
(a) Let's assume that Bernard answered x questions correctly. Since he answered all 20 questions, the number of questions answered incorrectly would be (20 - x).
The score for answering x questions correctly would be 6 points per question, so the total points earned for correct answers would be 6x.
The score for answering (20 - x) questions incorrectly would be -4 points per question, so the total points deducted for incorrect answers would be -4(20 - x).
Since the total score is zero, we can set up the equation:
6x - 4(20 - x) = 0
Solving this equation will give us the value of x, which represents the number of questions Bernard answered correctly.
(b) Let's assume that Raymond answered y questions correctly. Since the number of questions he answered incorrectly is the same as the number of questions answered correctly, the total number of questions answered correctly and incorrectly would be 2y.
The number of questions left blank can be found by subtracting the total answered questions (2y) from the total number of questions (20):
20 - 2y
The score for answering y questions correctly would be 6 points per question, so the total points earned for correct answers would be 6y.
The score for answering y questions incorrectly would be -4 points per question, so the total points deducted for incorrect answers would be -4y.
The score for leaving (20 - 2y) questions blank would be 0 points.
Since the total score is 16, we can set up the equation:
6y - 4y + 0 = 16
Solving this equation will give us the value of y, which represents the number of questions Raymond answered correctly, and also the number of questions he answered incorrectly.
Chrissy went to the store to buy some art supplies. She spent 1/5 of her money on paint and 1/3 of her money on paper. What fraction of her money did she spend?(1 point) Responses 23 2 thirds 8/15 8 over 15 7/15 7 over 15 2/15
To determine the fraction of Chrissy's money that she spent, we need to find a common denominator for the fractions 1/5 and 1/3. The least common multiple (LCM) of 5 and 3 is 15.
Now let's convert both fractions to have a denominator of 15:
1/5 = 3/15
1/3 = 5/15
Adding these fractions, we get:
3/15 + 5/15 = 8/15
Therefore, Chrissy spent 8/15 of her money.
James and Luke took a three-day hike along the Appalachian Trail. The first day they hiked 416 miles, the second day they hiked 323 miles, and the third day they hiked 423 miles. What total distance did the boys hike? You must type your response as a simplified mixed number. To type your fraction, click the box below. Type the whole number first. Then, select the fraction button where you see an x to the left of a fraction bar. Use your mouse or the up and down arrows on your keyboard to move between the numerator and denominator. The link below is a screen shot of the button you want to select.
To find the total distance James and Luke hiked, we need to add up the distances they hiked each day.
416 miles (first day) + 323 miles (second day) + 423 miles (third day) = 1162 miles.
Therefore, the boys hiked a total distance of 1162 miles.
James and Luke took a three-day hike along the Appalachian Trail. The first day they hiked 4 1/6 miles, the second day they hiked 3 2/3 miles, and the third day they hiked 4 2/3 miles. What total distance did the boys hike? You must type your response as a simplified mixed number. To type your fraction, click the box below. Type the whole number first. Then, select the fraction button where you see an x to the left of a fraction bar. Use your mouse or the up and down arrows on your keyboard to move between the numerator and denominator. The link below is a screen shot of the button you want to select.
To find the total distance James and Luke hiked, we need to add up the distances they hiked each day.
4 1/6 miles (first day) + 3 2/3 miles (second day) + 4 2/3 miles (third day)
Let's convert these mixed numbers to improper fractions:
4 1/6 = (4 * 6 + 1)/6 = 25/6
3 2/3 = (3 * 3 + 2)/3 = 11/3
4 2/3 = (4 * 3 + 2)/3 = 14/3
Now we can add these fractions:
25/6 + 11/3 + 14/3
To add these fractions, we need a common denominator, which is 6 in this case. So let's convert 11/3 and 14/3 to have a denominator of 6:
11/3 = (11 * 2)/ (3 * 2) = 22/6
14/3 = (14 * 2)/ (3 * 2) = 28/6
Now let's add these fractions:
25/6 + 22/6 + 28/6 = (25 + 22 + 28)/6 = 75/6
Since we need to express the answer as a simplified mixed number, let's divide the numerator by the denominator and express any remainder as a fraction:
75 รท 6 = 12 with a remainder of 3
So the total distance James and Luke hiked is 12 3/6 miles. To simplify this mixed number, we can reduce the fraction:
3/6 = 1/2
Therefore, the total distance they hiked is 12 1/2 miles.