Ratios in Data Sufficiency

We know that ratios are the building blocks for a lot of other concepts such as time/speed, work/rate and mixtures. As such, we spend a lot of time getting comfortable with understanding and manipulating ratios, so the questions that test ratios seem simple enough, but not always! Just like questions from all other test areas, questions on ratios can be tricky too, especially when they are formatted as Data Sufficiency questions. Let’s look at two cases today: when a little bit of data is sufficient, and when a lot of data is insufficient. When a little bit of data is sufficient! Three brothers shared all the proceeds from the sale of their inherited property. If the eldest brother received exactly 5/8 of the total proceeds, how much money did the youngest brother (who received the smallest share) receive from the sale? Statement 1: The youngest brother received exactly 1/5 the amount received by the middle brother. Statement 2: The middle brother received exactly half of the two million dollars received by the eldest brother. First impressions on reading this question? The question stem gives the fraction of money received by one brother. Statement 1 gives the fraction of money received by the youngest brother relative to the amount received by the middle brother. Statement 2 gives the fraction of money received by the middle brother relative to the eldest brother and an actual amount. It seems like the three of these together give us all the information we need. Let’s dig deeper now. From the Question stem: Eldest brother’s share = (5/8) of Total Statement 1: Youngest Brother’s share = (1/5) * Middle brother’s share We don’t have any actual number – all the information is in fraction/ratio form. Without an actual value, we cannot find the amount of money received by the youngest brother, therefore, Statement 1 alone is not sufficient. Statement 2: Middle brother’s share = (1/2) * Eldest brother’s share, and the eldest brother’s share = 2 million dollars Middle brother’s share = (1/2) * 2 million dollars = 1 million dollars Now, we might be tempted to jump to Statement 1 where the relation between youngest brother’s share and middle brother’s share is given, but hold on: we don’t need that information. We know from the question stem that the eldest brother’s share is (5/8) of the total share. So 2 million = (5/8) of the total share, therefore the total share = 3.2 million dollars. We already know the share of the eldest and middle brothers, so we can subtract their shares out of the total and get the share of the youngest brother. Youngest brother’s share = 3.2 million – 2 million – 1 million = 0.2 million dollars Statement 2 alone is sufficient, therefore, the answer is B. When a lot of data is insufficient! A department manager distributed a number of books, calendars, and diaries among the staff in the department, with each staff member receiving x books, y calendars, and z diaries. How many staff members were in the department? Statement 1: The numbers of books, calendars, and diaries that each staff member received were in the ratio 2:3:4, respectively. Statement 2: The manager distributed a total of 18 books, 27 calendars, and 36 diaries. First impressions on reading this question? The question stem tells us that each staff member received the same number of books, calendars, and diaries. Statement 1 gives us the ratio of books, calendars and diaries. Statement 2 gives us the actual numbers. It certainly seems that we should be able to obtain the answer. Let’s find out: Looking at the question stem, Staff Member 1 recieved x books, y calendars, and z diaries, Staff Member 2 recieved x books, y calendars, and z diaries… and so on until Staff Member n (who also recieves x books, y calendars, and z diaries). With this in mind, the total number of books = nx, the total number of calendars = ny, and the total number of diaries = nz. Question: What is n? Statement 1 tells us that x:y:z = 2:3:4. This means the values of x, y and z can be: 2, 3, and 4, or 4, 6, and 8, or 6, 9, and 12, or any other values in the ratio 2:3:4. They needn’t necessarily be 2, 3 and 4, they just need the required ratio of 2:3:4. Obviously, n can be anything here, therefore, Statement 1 alone is not sufficient. Statement 2 tell us that nx = 18, ny = 27, and nz = 36. Now we know the actual values of nx, ny and nz, but we still don’t know the values of x, y, z and n. They could be 2, 3, 4 and 9 or 6, 9, 12 and 3 Therefore, Statement 2 alone is also not sufficient. Considering both statements together, note that Statement 2 tells us that nx:ny:nz = 18:27:36 = 2:3:4 (they had 9 as a common factor). Since n is a common factor on left side, x:y:z = 2:3:4 (ratios are best expressed in the lowest form). This is a case of what we call “we already knew that” – information given in Statement 1 is already a part of Statement 2, so it is not possible that Statement 2 alone is not sufficient but that together Statement 1 and 2 are. Hence, both statements together are not sufficient, and our answer must be E. A question that arises often here is, “Why can’t we say that the number of staff members must be 9?” This is because the ratio of 2:3:4 is same as the ratio of 6:9:12, which is same as 18:27:36 (when you multiply each number of a ratio by the same number, the ratio remains unchanged). If 18 books, 27 calendars, and 36 diaries are distributed in the ratio 2:3:4, we could give them all to one person, or to 3 people (giving them each 6 books, 9 calendars and 12 diaries), or to 9 people (giving them each 2 books, 3 calendars and 4 diaries). When we see 18, 27 and 36, what comes to mind is that the number of people could have been 9, which would mean that the department manager distributed 2 books, 3 calendars and 4 diaries to each person. But we know that 9 is divisible by 3, which should remind us that the number of people could also be 3, which would mean that the manager distributed 6 books, 9 calendars and 12 diaries to each person. As such, we still don’t know how many staff members there are, and our answer remians E.