#instruction#
Review the above answer and determine whether it’s correct. If wrong, find the problems with it.
Finally, conclude with either '[[correct]]' if the above answer is correct or '[[wrong]]' if it is incorrect.
Here are some examples:
# ########## #
#Example#
---

Question: In the coordinate plane, points (x, 1) and (5, y) are on line k. If line k passes through the origin and has slope 1/5, then what are the values of x and y respectively?
Answer Choices: A)4 and 1 B)1 and 5 C)5 and 1 D)3 and 5 E)5 and 3
Answer: Since line k passes through the origin and has a slope of 1/5, it means that the line's equation is of the form y = (1/5)x.
For point (x, 1), we have y = 1, so we can set up the equation:
1 = (1/5)x
Now, solve for x:
x = 5
For point (5, y), we have x = 5, so we can set up the equation:
y = (1/5)(5)
Now, solve for y:
y = 1
So, the values of x and y are 5 and 1, respectively. The answer is C).
Verdict: The original solution starts by stating the equation of line k, given its slope is 1/5 and it passes through the origin. The equation of such a line is correctly identified as y = 1/5x.
For point (x, 1), the solution correctly substitutes y = 1 into the equation, leading to 1 = 1/5x. Solving for x results in x = 5, which is correctly determined.
For point (5, y), the solution appropriately substitutes x = 5 into the line's equation, yielding y = 1/5 × 5.
Simplifying this gives y=1, which is also correct.
The solution methodically applies the concept of a line's equation to find the values of x and y for the given points. The calculations are accurate, and the final answer, identifying the values of x and y as 5 and 1 respectively (corresponding to answer choice C), is correct.
In conclusion, the analysis of the provided answer reveals that it is methodologically sound and accurately solves the problem.
So the verdict is: [[correct]]

---

Question: Suppose a, b, and c are positive integers with a < b < c such that 1/a + 1/b + 1/c = 1. What is a + b + c?
Answer Choices: A)1 B)4 C)9 D)11 E)no such integers exist
Answer: This is a classic problem involving fractions and positive integers. You can approach it as follows:
Given: 1/a + 1/b + 1/c = 1
To find a + b + c, you can start by simplifying the equation:
1/a + 1/b + 1/c = 1
Now, let's find a common denominator for the fractions on the left side of the equation, which is abc:
(bc + ac + ab) / (abc) = 1
Now, you have:
bc + ac + ab = abc
Divide both sides by abc to isolate the terms:
1/c + 1/b + 1/a = 1
Now, you can see that a, b, and c are positive integers, and they must be greater than 1 because 1/c, 1/b, and 1/a are all positive fractions. Therefore, a, b, and c must be at least 2.
If a, b, and c are all equal to 2, the equation becomes:
1/2 + 1/2 + 1/2 = 1
So, in this case, a + b + c = 2 + 2 + 2 = 6.
Therefore, the answer is not "no such integers exist," and the correct answer is C) 6.
The answer is C)
Verdict:
The original solution to this problem starts by restating the given equation: 1/a + 1/b + 1/c = 1. The approach taken is to find a common denominator and then simplify the equation. This process is correctly executed, resulting in the equation bc + ac + ab = abc.
However, the solution goes wrong in the interpretation of this equation. It incorrectly states that dividing both sides by abc gives 1/c + 1/b + 1/a = 1, which is just a restatement of the original equation and doesn't progress towards a solution. Moreover, the conclusion that a, b, and c must all be at least 2 is unfounded.
The solution then incorrectly assumes that a, b, and c can all be equal, which contradicts the problem's stipulation that a < b < c. The assumption that a = b = c = 2 leads to an incorrect solution of a + b + c = 6, which is not even an option among the answer choices.
So the verdict is: [[wrong]]

---

Question: A question paper consists of three sections 4,5 and 6 questions respectively. Attempting one question from each section is compulsory but a candidate need not attempt all the questions. In how many ways can a candidate attempt the questions?
Answer Choices: A)119 B)(4!-1)*(5!-1)*(6!-1) C)209 D)29295 E)None
Answer: The number of ways a candidate can attempt the questions is equal to the product of the number of choices for each section.
Section 1 has 4 questions, and the candidate must attempt one of them, so there are 4 choices for the first section.
Section 2 has 5 questions, and the candidate must attempt one of them, so there are 5 choices for the second section.
Section 3 has 6 questions, and the candidate must attempt one of them, so there are 6 choices for the third section.
To find the total number of ways, simply multiply the choices for each section: 4 (choices for section 1) * 5 (choices for section 2) * 6 (choices for section 3) = 4 * 5 * 6 = 120 ways.
The answer is A).
Verdict: The original solution starts by correctly stating that the number of ways to attempt the questions is the product of the number of choices in each section. However, there is a critical mistake in the calculation.
Section 1 has 4 questions, and attempting one is compulsory. The candidate can attempt 1, 2, 3, or all 4 questions. The number of ways to choose questions from Section 1 is therefore 2^4 - 1 (all possible combinations minus the option of attempting none).
Similarly, for Section 2 with 5 questions, the number of ways is 2^5 - 1
For Section 3 with 6 questions, the number of ways is 2^6 - 1.
The total number of ways is thus the product of these individual choices:
(2^4 - 1) × (2^5 - 1) × (2^6 - 1)
Calculating this, we get:
(16 - 1) × (32 - 1) × (64 -1) = 15 × 31 × 63 = 29295
The solution incorrectly calculates the total number of ways as 4 * 5 * 6 = 120, which is a simplistic approach not considering the option to attempt more than one question per section.
So the verdict is: [[wrong]]

---

Question: A person can walk at a constant rate of 8mph and can bike at a rate of 16mph. If he wants to travel 64 miles in 8 hours using bike and walking at their constant rates, how much distance would he require to walk?
Answer Choices: A)20 B)30 C)48 D)64 E)72
Answer: Let's denote the distance the person walks as "x" miles and the distance he bikes as "y" miles. We have two pieces of information:
He wants to travel a total of 64 miles: x + y = 64
He can walk at a rate of 8 mph and bike at a rate of 16 mph, and he has 8 hours to complete the journey: The time it takes to walk (x/8) plus the time it takes to bike (y/16) equals 8 hours: (x/8) + (y/16) = 8
Now we can solve this system of equations for x and y:
From the first equation, we can express y as y = 64 - x.
Substitute this expression for y into the second equation:
(x/8) + ((64 - x)/16) = 8
Now, let's simplify this equation:
Multiply both sides of the equation by 16 to get rid of the fractions:
2x + (64 - x) = 128
Now, simplify further:
2x + 64 - x = 128
Combine like terms:
x + 64 = 128
Subtract 64 from both sides:
x = 128 - 64
x = 64
So, the person would need to walk 64 miles. The answer is D).
Verdict:
The original solution starts by setting up a system of equations based on the information given: the person walks "x" miles and bikes "y" miles, totaling 64 miles, and spends a total of 8 hours doing both activities at their respective speeds.
The first equation is x + y = 64, which is correct.
The second equation is x/8 + y/16 = 8. This equation correctly represents the time spent walking and biking.
The solution then correctly substitutes y from the first equation into the second equation, resulting in x/8 + (64-x)/16 =8. The algebraic steps taken to solve for x are as follows:
Multiply through by 16: 2x + 64 - x = 128
Simplify: x+64=128
Solve for x: x=128−64, which gives x=64.
So the verdict is: [[correct]]

---

# ########## #
#Question#
Please consider the example provided and think it step by step.
Question: {question}
Answer: {solution}
Verdict:
# ########## #
#Answer Format#
[[correct/wrong]]
# ########## #