Sets and Ratios

Let’s discuss Quant questions on two different topics of discussion – sets and ratios. Both questions are official and of higher difficulty. Question 1: Of the 1400 college teachers surveyed, 42% said they considered engaging in research an essential goal. How many of the college teacher surveyed were women? Statement I: In the survey 36% of men and 50% of women said that they consider engaging in research activity an essential goal. Statement II: In the survey 288 men said that they consider engaging in research activity an essential goal. Solution: On reading the question stem we realise that this question involves two variables: Research Essential – Not Essential Men – Women This should immediately make us think about a matrix. Not that we cannot solve the question without one, but you know that I am a huge proponent of visual approaches. We are given that 42% of total teachers (1400) considered research essential. So this means that 58% did not consider it essential. No need to actually calculate the number right now, let’s wait and see what else we know (anyway, we love to procrastinate calculations in Data Sufficiency questions). Statement I: In the survey 36% of men and 50% of women said that they consider engaging in research activity an essential goal. Say the number of women is W. We need the value of W. The number of men must be ‘Total – W’ = 1400 – W. 36% of men and 50% of women consider research essential. Knowing this, we see that we get: 36% * (1400 – W) + 50% * W = 42% * 1400 This is a linear equation in W so we can solve it to get the value of W. Therefore, this statement alone is sufficient. Statement II: In the survey 288 men said that they consider engaging in research activity an essential goal. This statement doesn’t tell us the number of women who consider research essential, so it is not sufficient alone, therefore the answer is A, Statement I alone is sufficient but Statement II is not. Question 2: If the ratio of the number of teachers to the number of students is the same in School District A and School District B, what is the ratio of the number of students in School District A to the number of students in School District B? Statement I: There are 10,000 more students in School District A than there are in School District B. Statement II: The ratio of the number of teachers to the number of students in School District A is 1 to 20. Solution: In both schools, the ratio of the number of teachers : the number of students is the same. Statement I: There are 10,000 more students in School District A than there are in School District B. We don’t know the number of students in either school district, so it is not informative enough to know that School District A has 10,000 more students. Therefore, this statement alone is not sufficient. Statement II: The ratio of the number of teachers to the number of students in School District A is 1 to 20. With this statement, we know that the ratio of the number of teachers : the number of students in School District A = 1:20. Say the number of teachers in A = a; the number of students in A = 20a. We also know the ratio of the number of teachers : the number of students in School District B = 1:20. Say the number of teachers in B = b; the number of students in B = 20b. Mind you, we don’t know the value of a and b. All we know is that the teacher student ratio is 1:20 in both. The ratio of the number of students in A: the number of students in B = 20a : 20b = a:b. With this ratio, we don’t know a:b (even using both statements, we just know that a – b = 10,000). Therefore, the answer is E, Statements 1 and 2 together are not sufficient. Were you able to solve both questions effortlessly? No? Don’t worry, that’s what we are here for! (Ignore the preposition at the end. It sounds most natural this way.)