GMAT: Data sufficiency 14

We're on problem 63. Carlotta can drive from her home to her office by one of two possible routes. If she must also return by one of these routes, what is the distance of the shorter route? So essentially she has a long route to go to her office, that's her home, that's her office. And then she has a short route, with fewer curves, that's my guess. And she can go and come back by one of two-- Statement number one: when she drives from her home to her office by the shorter route and returns by the longer route, she drives a total of 42 kilometers. So when she goes this, way shorter route this way and then longer route back, it's 42 kilometers. So shorter plus longer is equal to 42. And that by itself doesn't let me know what the distance of the shorter route is. Statement number two: when she drives both ways from her home to her office and back by the longer route, she drives a total of 46 kilometers. So if she goes by the longer route and then by the longer route again, so 2 times a longer route is equal to 46 kilometers. Well, this by itself helps me figure out what the longer route is, but they want to know the distance of the shorter route. So this, I could just eyeball this, the longer route is 23 kilometers. But if you want to know the distance of the shorter route you would need this information as well. The shorter route plus 23 is equal to 42. So the shorter route is 19 kilometers. So you need both of the statements. Both statements are sufficient, but one by themselves are not enough to figure out what the shorter route is. If they ask for the longer route, just statement two could have been good enough. Next question. 64. Is x greater than y? My computer's updating something. It's a little bit-- Is x greater than y? Statement number one tells us that x is equal to y plus 2. Well, let me think about that. No matter what y is, x is going to be 2 more. Even if y is a negative million, x is still going to be a bigger number. It's going to be 2 more than negative a million. If it's y is negative 10, this will be negative 8. No matter what x will be greater than y. So statement one is sufficient. Statement number two: x/2 is equal to y minus 1. OK. This doesn't look as clear there. So x is equal to 2y minus 2. So we just have to come up with a condition, one y where y is less than x and another y where y is greater than x and then we would have proven our point. So let's think about it. When y is 0-- this is using statement number two-- when y is 0, what's x? x is minus 2. So that's a statement where y is greater than x. And then when y is equal to 10, what's x? x is 18. So this is a statement where y is less than x. So statement number two actually tells us nothing about whether x is greater than y or y is greater than x. So statement number two is useless. Statement number one, alone, is sufficient. Next problem. 65. If m is an integer is m odd? Statement number one: m divided by 2 not-- they write it real big and bold-- not even integer. Well, that doesn't tell me much. I mean that doesn't tell me that this is odd. In fact, it's not an even integer but if this was an odd integer then that means that m is an even integer, right? Anything that's divisible by 2 and results in an integer-- right, that's essentially saying that it's divisible by 2-- by definition is going to be even. So for example, what if m divided by 2 is equal to 3, right? 3 is not an even integer but still m would equal 6. And then m would be even. And I mean they're not telling us much here. m divided by 2 could be equal to 2.5 in which case m would be equal to 5 so that would be an odd integer. So this by itself doesn't give me much information. Statement number two: m minus 3 is an even integer. Well, this is useful, right? If you subtract an odd number from a number and you get an even number, this number is going to be even. And let's think about that. If I have an even number-- let me just write a bunch of numbers. Let me write x, y, and z. And let's say they're consecutive, right? Actually let me write it even better. Let's say that x is even-- let's say x is odd. Then x plus 1 is going to be what? It's going to be even, right? The next number will be even. Then x plus 2 is going to be odd. And then x plus 3 is going to be even. And then if you go the other way-- actually I should have gone to the way first-- x minus 1 would be even. x minus 2 would be odd. And x minus 3 would be even, right? If you have an odd number and you subtract out an odd number and you get an even number then this number is going to be odd. So this statement two by itself is actually sufficient. And play around with some numbers if that intuition doesn't make sense. Statement two, by itself, is sufficient to solve this problem and you don't need statement one at all. Next problem. 66. what is the area of the triangular region ABC above? Well, they've drawn us a little triangle so I will draw us a triangle. Let's see, we have our base and then, it looks pretty symmetric. I'll try to draw it the way they drew it. One end, that's the other end, and then this is the altitude. Good enough. And then they label it A, B, C, D. This is a right angle. This is x degrees. Alright, what is the area of the triangular region? So area of a triangle is base times height times 1/2. So if we can do that then we're all done. One: the product of BD and AC is 20. So BD times AC-- oh no, sorry, BD times AC, oh that's it. So BD is the height. So area, let me just write it, area is equal to 1/2 times this altitude, BD, times the base, times AC, right? Statement number one, they tell us what BD times AC is. They tell us that BD times AC is equal to 20. So that immediately follows, if this thing is equal to 20 then the area is equal to 10. And you didn't even have to figure that out. So statement number one, by itself, is sufficient. Statement number two: x is equal to 45 degrees. This one's a little bit more interesting. x is equal to 45 degrees. Well, I mean we could do a little angle game, we could say, OK, this is a 90 degree angle and we could say if this is 90, this is 45, and this is going to be 45-- actually that's about where we could stop. And this tells us no information about what any of the sides are so you could imagine if this is 45 degrees, but this could be a million mile high triangle or it could be a nanometer high triangle. We don't know. It could be of any size. You could scale it up or down. This just tells us the degree and the angle information. So two, by itself, is useless. So for this one, statement one, alone, is sufficient to answer this problem, A. Next problem. 67. What is the value of b plus c? Alright, number one: they tell us that ab plus cd plus ac plus bd is equal to 6. And this by itself doesn't help us much because there's all these a's and d's here and they could be anything. So we can't solve for b plus c. But I'm already suspecting that we could factor out the a's and d's somewhere. So we can factor out the b's plus c's, then we might be able to figure out the numbers if we're given more information about a and d. Statement two, and they give us more information about a and d. a plus d is equal to 4. So let me see if I can simplify this top one. Let me see if I can factor out a bunch of b plus c's. So if we think about it, let me think. So if you take this term and that term and you factor out the a you get a-- let me do it in a different color-- so this term and that term is equal to a times what? b plus c, right? You can multiply that out and you get ab plus ac. And then plus, and let's do the same thing with this term, d. That's d times c plus b. But we can switch the order, b plus c-- and if you don't believe me that this equals this, multiply it out. All I did is I grouped the ab and the ac terms and then factored out the a and then I grouped the cd and the bd terms and factored out the d. And then now we can factor out a b plus c. We can do the reverse distributive property again. So now this equals a plus d times b plus c, right? And if you don't believe me, just multiply b plus c times a and then b plus c times d and you get that, right? That's just the distributive property. But now we're ready to solve because they told us that this thing's equal to 6 and they tell us that this thing is equal to 4. So b plus c is going to be equal to 6/4 or 1 and 1/2. And we're done. You need both statements together. Both statements, together, are sufficient to solve this problem.