Guests at a recent party ate a total of fifteen hamburgers. Each guest who was neither a student nor a vegetarian ate exactly one hamburger. No hamburger was eaten by any guest who was a student, a vegetarian, or both. If half of the guests were vegetarians, how many guests attended the party? (1) The vegetarians attended the party at a rate of 2 students to every 3 non-students, half the rate for non-vegetarians. (2) 30% of the guests were vegetarian non-students.
Solution:
We have two categories of people: vegetarians and students. Like any overlapping sets question, people can be in these groups or not in these groups, so there aren't actually 4 categories to track. We know that there are a total of 15 hamburgers; we know that anyone neither a vegetarian nor a student ate 1 hamburger each; we know that anyone who's a vegetarian, student or both ate no hamburgers. Accordingly, we know there must be exactly 15 people (15 burgers, 1 burger per person) who are neither vegetarians nor students. We're also told that half of the guests are vegetarians; therefore, half the guests are non-vegetarians. Q: how many guests were at the party. Well, we know that: Total = veg + non veg or Total = 2(veg) So, if we can figure out the number of vegetarians, we can calculate the total number of guests. (1) The vegetarians attended the party at a rate of 2 students to every 3 non-students, half the rate for non-vegetarians. This sounds very complicated. Fortunately, this is data sufficiency, so we just need to understand the kind of information we have, rather than all the details of the information. We're given the ratios of student vegetarians to non-student vegetarians and the rates for student non-vegetarians to non-student non-vegetarians. Since we know that vegetarians = non-vegetarians, we can now use these ratios to calculate the ratios of all the different types of guests. Accordingly, we know what portion of the guests are non-student + non-vegetarian. We also know there are 15 of these silly people. With a part-to-whole ratio and the number that goes along with the part, we can calculate the whole: Sufficient. (2) 30% of the guests were vegetarian non-students. We know that 50% of the guests are vegetarians, so we now know that 20% of the guests were vegetarian students. However, we still don't know what % of the guests are students, so there's no way to figure out how "15" relates to total guests: Insufficient. (1) is sufficient, (2) isn't... choose (A).
Each day after an item is lost the probability of finding that item is halved. If 3 days after a certain item is lost the probability 1/64, what was the initial probability of finding the item? a)1/32 b)1/8 c)1/4 d)1/2 e)1
Solution:
If we let the original probability be x, we get: x(1/2)(1/2)(1/2) = 1/64 x(1/8) = 1/64 x = (1/64)(8/1) = 8/64 = 1/8. Choose (B).
Is mp greater than m? (1) m > p > 0 (2) p is less than 1
Solution:
Let's start at the beginning: Is mp > m? If we want to rewrite this safely, we subtract m from both sides, to get: is mp - m > 0 and then factor out m: is m(p-1) > 0 Now we ask ourselves, when is a product of two terms greater than 0? We answer ourselves: when both terms have the same sign. So, to get a yes answer, either: m>0 and p-1>0 (i.e. p>1) OR m<0 and p-1<0 (i.e. p<1) Now let's look at the statements: (1) m > p > 0 we know that m>0, but do we know if p>1? No! So, (p-1) could be positive or negative: insufficient. (2) P is less than 1 No info about m: insufficient. From (1) we know that m is positive. From (1) and (2) together, we know that p is a positive fraction. If p is a positive fraction, then (p-1) will always be negative. If m is positive and p-1 is negative, the question changes from: Is m(p-1) > 0 to: is (+)*(-) > 0? To which the answer is "DEFINITELY NOT": sufficient, choose (C).
Solution:
We have two categories of people: vegetarians and students. Like any overlapping sets question, people can be in these groups or not in these groups, so there aren't actually 4 categories to track. We know that there are a total of 15 hamburgers; we know that anyone neither a vegetarian nor a student ate 1 hamburger each; we know that anyone who's a vegetarian, student or both ate no hamburgers. Accordingly, we know there must be exactly 15 people (15 burgers, 1 burger per person) who are neither vegetarians nor students. We're also told that half of the guests are vegetarians; therefore, half the guests are non-vegetarians. Q: how many guests were at the party. Well, we know that: Total = veg + non veg or Total = 2(veg) So, if we can figure out the number of vegetarians, we can calculate the total number of guests. (1) The vegetarians attended the party at a rate of 2 students to every 3 non-students, half the rate for non-vegetarians. This sounds very complicated. Fortunately, this is data sufficiency, so we just need to understand the kind of information we have, rather than all the details of the information. We're given the ratios of student vegetarians to non-student vegetarians and the rates for student non-vegetarians to non-student non-vegetarians. Since we know that vegetarians = non-vegetarians, we can now use these ratios to calculate the ratios of all the different types of guests. Accordingly, we know what portion of the guests are non-student + non-vegetarian. We also know there are 15 of these silly people. With a part-to-whole ratio and the number that goes along with the part, we can calculate the whole: Sufficient. (2) 30% of the guests were vegetarian non-students. We know that 50% of the guests are vegetarians, so we now know that 20% of the guests were vegetarian students. However, we still don't know what % of the guests are students, so there's no way to figure out how "15" relates to total guests: Insufficient. (1) is sufficient, (2) isn't... choose (A).
Each day after an item is lost the probability of finding that item is halved. If 3 days after a certain item is lost the probability 1/64, what was the initial probability of finding the item? a)1/32 b)1/8 c)1/4 d)1/2 e)1
Solution:
If we let the original probability be x, we get: x(1/2)(1/2)(1/2) = 1/64 x(1/8) = 1/64 x = (1/64)(8/1) = 8/64 = 1/8. Choose (B).
Is mp greater than m? (1) m > p > 0 (2) p is less than 1
Solution:
Let's start at the beginning: Is mp > m? If we want to rewrite this safely, we subtract m from both sides, to get: is mp - m > 0 and then factor out m: is m(p-1) > 0 Now we ask ourselves, when is a product of two terms greater than 0? We answer ourselves: when both terms have the same sign. So, to get a yes answer, either: m>0 and p-1>0 (i.e. p>1) OR m<0 and p-1<0 (i.e. p<1) Now let's look at the statements: (1) m > p > 0 we know that m>0, but do we know if p>1? No! So, (p-1) could be positive or negative: insufficient. (2) P is less than 1 No info about m: insufficient. From (1) we know that m is positive. From (1) and (2) together, we know that p is a positive fraction. If p is a positive fraction, then (p-1) will always be negative. If m is positive and p-1 is negative, the question changes from: Is m(p-1) > 0 to: is (+)*(-) > 0? To which the answer is "DEFINITELY NOT": sufficient, choose (C).
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