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we're on problem 6363 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 could go and come back by one of the two with that kind of binary statement number one statement number one when she drives from her home to our office by the shorter route and returns by the longer route she drives a total of 42 kilometres so when she goes this way shorter out this way and then longer out back it's 42 kilometres 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 our office and back by the longer route she drives a total of 46 kilometres so she goes by the longer route and then by the longer route again so two times the longer route is equal to 46 kilometres well this by itself helps me figure out what the longer route is the way I want to know the distance of the shorter route so this you know I could just eyeball this it's the longer route is 23 kilometres but if you want to know the distance of the shorter route you would need this information as well and you know whatever the shorter route plus 23 is equal to 42 so it's what the shorter route is 19 kilometres 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 they ask for the longer route just statement two could have been good enough next question 64 is X greater than Y X greater than Y my computer is updating something it's a little bit is X greater than Y statement number one tells us that X is equal to y plus two well let me think about that no matter what Y is X is going to be two more even if Y is a negative a million X is still going to be a bigger number it's gonna be two more than negative a million if it's Y is negative ten this will be negative eight no matter what X will be greater than Y so this is all a statement one is sufficient statement number two statement number two x over two is equal to y -1 this doesn't look as clear there so X is equal to two Y minus two 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 zero when y is here this is using statement number two and y is zero what's X X is minus two so that's a statement where Y is greater than X and then when Y is I don't know when y is equal to 10 what's X 20 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 M statement number one statement number one M divided by two not they write a real big in bold not even integer not even integer well that doesn't tell me much I mean that doesn't tell me that 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 two and results in an integer right that's essentially saying this is divisible by two 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 and 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 then it 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 M minus 3 is even well this is useful right if if enough 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's let me just write a bunch of numbers let's see X 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 X is even let's say 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 or and then if you go the other way actually I should have gone the other way first X minus 1 would be even X minus 2 would be odd and X minus 3 would be even right when you subtract if you haven't if you have an odd number and you subtract out and if you subtract out an odd number and you get an even number then this number is going to be odd so this statement 2 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 if our base and then it looks it looks pretty symmetric I'll try to draw it the way they drew it it's one end 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 all right what is the area of the triangular region so area of a triangle is base times height times 1/2 so if we can figure if we can do that then we're all done 1 the product of BD and AC is 20 so BD so BD times AC I'm sorry BD times AC oh that's it so BD is the height so 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 1 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 you didn't have to figure that out so a statement number 1 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 you know do a little angle game we could say okay this is a ninety degree angle and we could say that 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 trial we don't know it could be of any size you could scale it up or down this just tells us the degree in the angle information so 2 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 B plus C all right number one they tell us that a B plus C D plus AC plus B D is equal to 6 and this file itself doesn't help us much as it felt these A's and d's here and we don't they could be anything so we can't solve for B plus C but I'm already suspecting that if we could factor out the A's and the D somewhere so we get so we can factor out the beef plus C's then we might be able to to figure out the numbers if we forgiven more information about a and D statement 2 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 different color so this term and that term is equal to a times what B plus C B plus C right you can multiply that out and you get a B plus a C 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 group a B in the AC terms and then factor out the a and then I grouped the CD and the BD terms and factored out the D and then now we can we can factor out a B plus C we can do the reverse just 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 put times D and you get that right that's just a distributive property but now we're ready to solve because they told us that this thing is equal to 6 so this thing is equal to 6 and they tell us that this thing is equal to 4 this is equal to 4 so B plus C is going to be equal to 6 over 4 or 1 and a half and we're done you need both statements together both statements together are sufficient to solve this problem