A store sells kurta only in small, medium, and large sizes, and only in red, yellow, and blue colors. Rupa buys exactly three kurta from the store.
- A shirt type consists of both a size and a color.
- Rupa does not buy two kurta of the same type.
- Rupa does not buy both a small shirt and a large shirt.
- No small red kurta are available.
- No large blue kurta are available.
- Which one of the following must be false?
- Two of the kurta that Rupa buys are small and two are red.
- Two of the kurta that Rupa buys are medium and two are red.
- Two of the kurta that Rupa buys are large and two are red.
- Two of the kurta that Rupa buys are small, one is yellow, and one is blue.
- Two of the kurta that Rupa buys are medium, one is yellow, and one is blue.
- If Rupa buys a small blue shirt, which one of the following must be false?
- Rupa buys two blue kurta.
- Rupa buys two red kurta.
- Rupa buys two yellow kurta.
- Rupa buys two small kurta.
- Rupa buys two medium kurta.
- If Rupa does not buy a medium yellow shirt, which one of the following must be true?
- Rupa buys either a medium red shirt or a small blue shirt.
- Rupa buys either a medium red shirt or a medium blue shirt.
- Rupa buys either a large red shirt or a small blue shirt.
- Rupa buys either a large red shirt or a medium red shirt.
- Rupa buys either a large yellow shirt or a small yellow shirt.
- If Rupa buys exactly one medium shirt and does not buy two kurta of the same color, then she cannot buy which one of the following?
- a medium red shirt
- a medium yellow shirt
- a medium blue shirt
- a large red shirt
- a large yellow shirt
- If neither large red kurta nor small blue kurta are available, which one of the following must Rupa buy?
- a red shirt
- a medium yellow shirt
- either a large shirt or a small shirt
- either a medium red shirt or a medium blue shirt
- either a large yellow shirt or a medium blue shirt
This set asks us to match shirt sizes—small, medium, and large—with shirt colors—red, yellow, and blue. Seems simple enough, but there’s a lot of ambiguity, at least at first, in the selection process. The Key Issues will deal with:
1) What types of kurta does Rupa buy?
2) What color of shirt can Rupa buy based on what other color and size of shirt is bought?
3) What size of shirt can Rupa buy based on what other color and size of shirt is bought?
Note the one bit of precision in this otherwise slippery set: Rupa buys exactly three kurta.
Let’s look the Rules over before discussing the Initial Setup, for convenience’s sake.
1) simply defines the word “type” in the context of the set, so that the term can be used freely in the rules and questions. We are to think of “type” as the size/color combination. Note that there are nine possible “types” (3 colors x 3 sizes): small red, small blue, small yellow, medium red, medium blue, etc.
2) Another loophole closer. It means that we must choose three different types out of the nine.
3) An injunction against ever picking small and large together. Just underline it for the moment.
4) One “type” is forbidden: small red. Think it through contrapositively: If a small shirt is chosen, it will have to be blue or yellow. And if a red shirt is chosen, it’ll have to be medium or large.
5) Here’s another forbidden “type” of shirt: large blue. Therefore, if a large shirt is chosen, it’ll have to be red or yellow. And if a blue shirt is chosen, it’ll have to be small or medium.
As matching sets go, this one is pretty loose, and Kaplan has identified three potential ways in which you might have dealt with it on the page. Take note of all three, for future reference. After all, what’s important is not this set, but what you can learn from this set for later work.
The Grid Approach. This involves setting up a 3 x 3 grid with the characteristics labeled, allowing you the potential to X any combinations rejected and √ the 3 that are chosen. It would look like so:
(Notice that we’ve gone ahead and indicated the upshot of Rules 4 and 5, the rules that toss out two of the nine possible shirt “types.”) As needed per question, you’d resketch the grid and enter its information, keeping in mind that you need to come up with exactly three √s at all times. Some students find such grids counterintuitive and find it difficult to take note, in the grid, of such injunctions as Rules 2 and 3. Those students may prefer
The Table Method—in which you simply enter the data through deduction:
This may be a more conventional way of working out the information on paper than the grid method. Here, you’d simply take note of the restrictions on the selection process and refer to them as you went along:
Finally, many students noticed that there is only a limited number of possibilities at work in terms of shirt size and resolved to use
The Work-It-All-Out Method. For starters, note that Rupa could conceivably buy one of each size, or all three of the same size:
But “sm med lg” violates Rule 3 (she can’t buy both small and large together). And “sm sm sm” is out, because it would mean buying one of each color, and she can’t buy small red (Rule 4); likewise “lg lg lg” is out, because she’d have to buy a large blue shirt which is forbidden (Rule 5). So the only possible size combination so far is “med med med”—one of each color.
Now if Rupa doesn’t buy one of each size or three of the same size, what’s left? Two of one size, and one of another. If other words,
But here again, Rule 3 comes into play rendering “sm sm lg” and “lg lg sm” impossible. In the end, there are only five possibilities for the sizes, and here they are, along with what we can deduce about color in each case:
med med med (obviously, one of each color)
sm sm med (one small blue, one small yellow, the medium’s color is uncertain)
med med sm (the small is blue or yellow, not sure about the two medium)
med med lg (the large is red or yellow, not sure about the two medium)
lg lg med (one large red, one large yellow, the medium is any color)
The advantage here is that so much is worked out in advance of the questions that the actual question work becomes a cinch. The disadvantage, of course, is that all of this work is time consuming, providing lots of opportunities to make mistakes or get discouraged.
Key Deductions: Clearly, the third method above works the most out in advance. Even using the grid or table methods, however, it’s possible to realize how limited Rupa’s choices really are and go into the questions with confidence. Also, we think it’s worth noticing how little we are told about the medium-sized kurta: As long as we never choose two of the same “type,” the medium-sized kurta offer the most freedom of color choice. But there’s one thing we do know for sure: No matter what, Rupa has to buy at least one medium shirt. Check it out. That’s a major realization worth making upfront.
Which one “must be false”—which one is impossible? Not much to do but try the answer choices; and the testmakers cut us a break, because the very first one, (A), is the one that’s a no go. If Rupa buys two small kurta, they must be yellow and blue (Rule 4). The third shirt can be red, either medium or large, but two red kurta are impossible. Here are combinations indicating that the other four choices are eminently possible:
(B) medium red, large red, medium yellow;
(C) medium red, large red, large yellow;
(D) small yellow, small blue, medium red;
(E) medium yellow, medium blue, large red.
If, as the stem says, Rupa buys a small blue, then Rule 3 means that all large kurta are out. And a glance at the choices tells us that we’re looking for what Rupa can’t buy two of. Since large kurta are out, then (Rule 4) only one red shirt (a medium one) can be purchased, and answer choice (B) is our answer. This, the active approach, is the fastest way to get the answer, but you could have tried out each choice until you came upon the right one.
(A) The stem has assigned Rupa a small blue shirt, and there’s nothing stopping her from buying a second blue one (in medium).
(C) Along with the small blue, she can buy two yellows of any size.
(D) and (E) Given that Rupa buys a small shirt, there are two possibilities: Either she buys two medium, in which case (E) is true, or she buys another small and one medium, in which case (D) is true.
All five of the possibilities we worked out (in the “Work-It-All-Out” discussion, above) include at least one medium shirt. With medium yellow forbidden, as it is here, then clearly that requisite medium has to be red or blue. And if you scanned the choices, you could have recognized that this is just what (B) is saying. Here are combinations demonstrating that the other choices needn’t be true:
(A) large red, large yellow, medium blue;
(C) medium red, small yellow, medium blue;
(D) small yellow, small blue, medium blue;
(E) medium red, large red, medium blue.
Given that only one medium shirt is purchased, the remaining kurta have to be two small or two large. (We worked all this out above.) If you proceed to match up the sizes with colors, remembering that in this question, each color is used exactly once, you can narrow the selection down to
(i) medium blue, large yellow, large red, or
(ii) medium red, small yellow, small blue.
Scanning the choices, notice that (B) is not included in either of our possibilities. And indeed, if Rupa buys a medium yellow shirt, either Rule 4 or 5 will be violated. As for the wrong choices, (A) is purchased in (ii) above, while (C), (D), and (E) are all purchased in (i).
Two more “types” are forbidden: large red and small blue, which narrows things down a lot. Rupa can still buy all three medium kurta (which eliminates choice (C), incidentally), and the only other possibilities are one small and two medium or one large and two medium.
Upshot: Rupa has to buy either two or three medium kurta, which brings us to answer choice (D). If she buys all three medium kurta, it’s obvious that she’ll have to buy either a medium red or medium blue (both, actually). But even if she buys only two medium, the only combinations are red and yellow, red and blue, or yellow and blue; in all possible combinations she will buy either a medium red or a medium blue (if not both). So (D)’s our answer.
(A) She needn’t buy red. She could buy only blue and yellow in some combination.
(B) She needn’t buy a medium yellow. She could buy a medium red, medium blue, and a yellow (small or large).
(C) See above. She could buy three medium.
(E) She can avoid purchasing either a large yellow or a medium blue shirt by going for small yellow, medium red, and medium yellow kurta.
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