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Current time:0:00Total duration:9:31

we're on problem 141 on Jane's credit card account the average daily balance for 30-day billing cycle it's good when I see these long paragraphs gives you a headache the average daily balance for 30-day billing cycle is the average of the daily balances at the end of each of the 30 days okay so they take the average balance of each of the days and and in the average it okay at the beginning of a certain 30-day billing cycle jane's credit card account had a balance of $600 at the beginning so so what Palmas is 141 so at the beginning she had a balance of $600 fair enough Jane made a payment of $300 on the account during the billing cycle so this is the beginning of the month and then at some point this is 30 days later at some point she made a $300 payment fair enough kenji if no other amounts were added or subtracted from the account during the billing cycle what was the average daily balance on Jane's account for the billing cycle okay so I think I'm visualizing this right if we say that this is days where this is day day zero and this is day 30 so essentially at the end of every day so at the end of day one so at the beginning of a certain billing cycle Jane's credit card account had a balance of 600 so we don't know what she might have paid it on day one so I'm not gonna put any labels here but she starts at 600 and that's her balance until some day beginning of Jane's on some day she pays off half of it and her balance goes down like that so the question is what is the average so it's going to be $600 times certain number of days plus $300 times a certain number of days divided by 30 so let me write that down the average is going to be equal to 600 times however many days she carried the 600 balance plus 300 times however many days she kept the 300 balance and that's going to be 30 minus X 30 minus X right if she kept the $600 if she paid after 15 days she would have had the $600 balance for 15 days and then she would have the 300 ollars for the remainder of the days if she paid after one day then he would have the $300 bounds for 29 days and all of that divided by 30 that's the average and I try to do that from the get-go because I just want to get kind of my algebraic brain around the problem so it becomes less abstract so statement number one statement number one let me keep that in the screen statement number one tells us Jaynes payment cycle was credited on the 21st day of the bailing cycle 21st day credited on the 21st day so that means that she had a 600 that X in this example that she had a $600 balance for 20 days and then on the 21st day her balance would have gone to 300 so the average is going to be equal to 600 she had a $600 balance for the first 20 days times 20 plus 300 times the remainder days right 30 so 300 for 10 days and that's all divided by the number of days so statement one alone is enough to figure out her average balance for the month statement two the average daily balance through the 25th day of the billing cycle was 540 dollars this is interesting so 540 is the average through the twenty-fifth day so if we average the first 25 days and the first 25 days so her balance was 600 for X days plus three hundred for the remainder it only in the first 25 days so it's X minus 25 so you can actually take this equation and solve for X it's actually a linear equation and I'm sure this time multiply both sides by 25 you get a number and then you can distribute this out at all the X terms all for X and then once you solve for X you can just use this equation up here to figure out the average daily balance for 30 days so actually each of these statements independently are sufficient to figure out this problem and actually was very critical that we kind of thought about it in this term from the get-go otherwise this would've been a very hard problem to get your get your hand around that is interesting I strangely really like that problem anyway next problem 142 142 if X is an integer X is an integer they're asking is 9 to the X plus 9 to the minus X equal to B who knows problem one says 3 to the X plus 3 to the minus X is equal to the square root of B plus 2 so right from the get-go I don't see I mean there's threes there's 9s there seems to be some relationship let's square both sides of this equation and see if it can reduce to something that's useful here so the right hand side Z's with the left hand side so you get three if you square it you get three to the 2x and then you get plus 2 times these multiplied by X you know the 3 to the X 3 to the minus X plus 3 to the minus 2x is equal to B plus 2 I just square both sides 3 to the X times 3 to the minus X that just equals 1 right you could add the exponents that's equal to 3 to the 0 that equals 1 or you could view that as 3 to the X divided by 3 to the X either way that equals 1 and then you're left with 2 on both sides of equation so you can get rid of that so then we're left with 3 to the 2x plus 3 to the minus 2x is equal to B but then we could rewrite this think about this 3 to the 2x that's the same thing as 3 squared to the X and this is the same thing as 3 to the -2 to the X is equal to B and I think now your bells are ringing in your head 3 squared that's the same thing as 9 to the X plus so I should do 3 squared to the minus X that's easier plus 9 to the minus X is equal to B so statement number 1 actually reduced to what we were trying to prove so statement number 1 at least alone is sufficient let's see what statement number two gets us statement number two says X is greater than 0 is equal to B X is greater than 0 is equal to B let's see we the original was 9 to the X plus 9 to the minus X is equal to B so they're now saying that B is equal to 0 so how does that help us so if B is equal to 0 let's subtract 9 I don't know 9 to the minus X from both sides you get 9 to the X is equal to minus 9 to the minus X now let me multiply let me get multiplied both sides by I don't know negative 9 to the X so then you get C I just want to make this minus 9 to the x times minus 9 to the X I'm just trying to simplify it so what do you have here the minus sign minus 9 to the 2x all right just add the exponents is equal to minus times a minus is a positive and then a minus X and a positive x add them together you get 0 so 9 0 is 1 so that simplifies to minus 9 to the 2x is equal to 1 minus 9 to the 2x is equal to 1 so Ken can this ever be true too can this ever be true let me think about that the only way something can something that's non 1 or non-negative 1 the only way that when you raise it to a power you can get a one is if you raise it to a 0th power so this is only true this is only true if X is equal to zero but they tell us that X is greater than zero so actually this is an interesting case using the information they gave us in problem number two using the information they gave us in problem number two I just want to make sure I'm I'm not missing so statement 1 was sufficient can we prove what we don't have enough information to prove the statement we have enough information to disprove the statement but not to prove it so I'm gonna stick with a I was going to say that we can say whether the statement is true or false but we definitely don't have enough information which a statement be to prove and I'm right so they want to know right you have to be able to prove the statement is true statement B the statement - actually proves that it is false it doesn't prove that it's true it answers the question but says no so statement one alone is sufficient to answer to say that this that that statement is true and I'll continue in the next video