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problem 7777 if miss Smith's income was 20% more for her 99 1991 than it was for 1990 how much was her income in 1991 so 20% more in 1949 t91 so ninth income let's call it income 1991 is 20% more than then 1990 so that means 1.2 times income in 1990 no Pia you get that right if something is 20% more than something else is going to be 1.2 times that right 12 is 20% more than 10 and 12 is 1.2 times 10 right or you could view it as income plus 20% of income which is 1.2 times income either way so let's see what they tell us what do we have to figure out how much was okay where we're trying to figure out this income in 1991 statement number one miss Smith's income for the first six months of 1990 was seventeen thousand five hundred and the income for the last six months of 1990 was twenty thousand dollars what seems so is this youth they're selling that she made seventeen thousand five hundred in the first six months of 1990 and her income for the last six months of 1990 was 20 or twenty thousand well they're essentially telling us the total income for 1990 right the first six months in the last six months or 12 months in a year so her total income for 1990 was thirty seven thousand five hundred that equals income for 1990 so clearly if we know this is this we just multiply that times 1.2 and we get the income for 1991 so this statement alone is sufficient let's see what they give us for statement number two misfits income for 1991 was 7500 greater than for 1990 so they say income of 1991 was 7500 so it equals income for 1990 plus seventy five hundred plus seventy five hundred well this alone well this alone does help us because they already gave it given us this so we have two linear equations right this is one linear equation and two unknowns this is another linear equation in two unknowns so we have two linear equations and two unknowns we can solve this probably the easiest ways is to substitute you could depending what you want to solve for but we've done that multiple times you could substitute one point two times nineteen ninety here and then solve for it or you can do the other way you could do divide by one point two here and then substitute it there but either way this is trivial algebra hopefully by this point to solve but this and this is definitely enough information to solve the problem so two equations with two unknowns and you can do that in in your spare time if you don't believe me so both statements alone are sufficient for this one seventy eight seventy eight and the figure above so I think I have to draw I mean the y-axis that's the x-axis and then they have a line see what I can do line looks something like that and then they tell us what do they tell us this is y this is X and they say this is P this right here is Q and then they draw this they call this point right here R and they drop that's like that and this Q is at Point C D and P is at Point a B and then they say in the figure above segments P R and Q R so P R and Q R so let me draw that out a little bit better this P R and Q R are each parallel to one of the rectangular coordinate axes okay fair enough this is parallel to the y-axis PR is parallel to the x-axis fair enough is the ratio of the length of QR to PR equal to one so the ratio of QR the PR is equal to one so they want to know QR over PR is that equal to one and immediately this should trigger something from Algebra one they're asking you essentially is the slope of this line equal to one right is the ratio of QR to PR so rise over run is the slope of this line equal to one so let's see let's see what we can do and slope is just you know you change in Y over change in X and well you know this is changing so what's this point first of all yes you don't have to know anything about slope I don't want to make you feel like you had to memorize some formulas it's all what's this point gonna be so it's going to be actually let's let's do it even better what is what is the length of QR going to be this is what I haven't looked at any of the data points right now what is length of QR well it's going to be this height so it's this Y which is d minus this Y this Y is going to be B right because all of this is y is equal to B right here so QR is going to be equal to D minus B and PR is the length on the x-axis it's going to be it's going to be this X right what is this X well this X is right here see X is equal to C it's going to be this X minus this X well here X is equal to a and so the ratio is equal to D minus B over C minus a which is that if you remember the formula for the slope of a line you just take the y1 minus y2 over x2 minus sorry Y 1 minus y 2 over X 1 minus X 2 but we just didn't have to memorize that it's intuition at this point right here our is the Point C the x-coordinate is C and the y-coordinate is B so hopefully that gives you the intuition now let's look at the statements you wouldn't have to do that on the real GMAT that would all be time statement number one tells us C is equal to three and D is equal to four so that by itself that just gives us the first part of this that doesn't help us to figure out this entire ratio so this by itself isn't that useful maybe in conjunction with what else they give us statement to a is equal to -2 and B is equal to minus 1 well if you use both of these statements together then we have everything here we have D we have B we have C and we have a so we can solve it so as both statements together are sufficient for solving for for knowing whether the ratio of Q R to P R is 1 or since she is the slope of this line equal to 1 next problem 79 79 while on a straight road car X and car y are traveling at different constant rates if car X is now one mile ahead of car y how many minutes from now will car X beat two miles ahead of car y so right now let me see let me see if I can so X X is here Y is here I think we had constant rates one mile so they've been traveling for some amount of time and X is one mile ahead and they're saying how long is it going to be before X is how many minutes before X is two miles ahead and there are constant rates so if they started off let's just think about if they started off at the same point and it took 10 minutes for X to get one mile ahead it would take another 10 minutes for to get 2 miles ahead well that's how I'm thinking about let's see what they give us for the for the statements statement number 1 car X is traveling at 50 miles per hour and car Y is traveling at 40 miles per hour well that's that seems to be pretty good information car X 50 miles per hour car Y it's 40 miles per hour so essentially car y is moving away from car exit 1 it's moving away 50 - forty miles per hour so if from car at wise point-of-view car X is always pulling away at ten miles per hour right does that make sense if car actual is going at 40 miles per hour they wouldn't he wouldn't be pulling away at all if it was going at 41 miles per hour would be pulling away in a incremental 1 mile per hour and you know as long as we're not approaching the speed of light we can assume Newtonian classical physics and we could just take the difference between the two so how long does it take for it to pull away another another mile well how many minutes if you're going 10 miles per hour relative to something else how many minutes does it take to go a mile well one you know you can figure that out but let me figure that out for you so you know distance is equal to rate times time so if your distance you want to know is one mile and your rate is equal to 10 miles per hour 10 miles per hour times time what's the time going to be equal time is going to be equal to 1 1/10 of an hour or 6 minutes so that's the answer number one one alone is sufficient or the answer number 79 one alone is sufficient let's see what they give us for number two statement two three minutes ago car X was one half mile ahead of car Y okay so three minutes ago the state of affairs was this Y was here X was here and it was a half mile difference so what does that tell us that's actually pretty good information - three minutes ago car X was half mile ahead of car y now car X is one mile ahead so in three minutes in three minutes X X pull you could say goes pulls away by three pulls away by half a mile right and they're going at constant velocity so they're the relative velocities between the two don't change so if it takes three minutes for X to pull away by half a mile it would take six minutes Forex to pull away by a mile right just multiplying by two there they're all going at the same constant velocities so six minutes ex pulls away pulls away by one mile and that's actually what they're asking because they say how many more minutes does it take X to pull away by another mile and they've probably been traveling for six minutes already and then another six minutes X would pull away by another mile so two alone is also sufficient so each of them independently are good enough to answer this question see in the next video