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# Linear models word problem: book

## Video transcript

naolia read a book cover-to-cover in a single session at a rate of 55 pages per hour 55 pages per hour after reading for four hours after reading for four hours he had 330 pages left to read how long is the book in pages and how long did it take neoya to read the entire book well there's a bunch of ways that we could approach to say like always pause the video and try to figure it out but I will go with one way first and then maybe I'll tackle it in another way so one way is let's set up a little table here and on one column let's think about let's think about how long naolia has read in hours so it's called that time T for time in hours this is how long he is read and in this column this is how much he has left to read so let's just write that L left to read and then this is going to be given in pages this is going to be given in pages so let's see we don't know at 0 hours we don't know how many pages he has left to read at if we knew that that's going to 0 hours before he's read that the pages left to read this could be the entire book this so that that would answer this question how long is the book so this is going to be interesting we don't know what happens at one hour after three two hours after three hours but they do tell us that after reading for four hours after reading for four hours he had 330 pages left to read so after four hours he had 330 pages 330 pages left to read so based on this and the fact that he's reading at 55 pages per hour can we back up to figure out all of this well remember every hour that goes by every hour that goes by he is read 55 pages so if he's read 55 pages remember this 330 this isn't how many pages he's read this is how many he has left to read so after every hour he has 55 fewer pages to read so three hours he should have 55 more pages to read so this should be at three hours 330 plus 55 would be 385 now let's see whether this makes sense so when our change in time this triangle is just the Greek letter Delta means change in when our change in time is plus one plus one our our change in pages left to read is going to be equal to negative fifty-five pages and that makes sense the pages left to read goes down every hour this is we're measuring not how much he's read we're measuring how much he has left to read so that should go down by fifty five pages every hour or if we were to go backwards through time it should go up so two hours he should have fifty five more pages to read so what's 3/8 what's 385 plus fifty five will see 385 plus five is 390 plus 50 is 440 so he'd have 440 pages and all I did is I added 55 and then after reading for one hour he would have 55 more pages than after reading for two hours so 440 plus 55 is 495 and then before he started reading right when he started reading he would have had to read even 55 more pages because after one hour he would have read those 55 pages so 495 + 55 is going to be let's see it's going to be add five you get to 500 plus another 50 is 550 pages so time equals zero he had 550 pages to read so that's how long the book is but how long does it take naolia to read the entire book well we could keep going we could say okay at the fifth hour this thing's going to go down by 55 so let's see if this goes down by 50 if this goes down by 50 we're going to get to 280 but then you go down five more it's going to go to 275 and we could keep going on and on and on or we could just say look he's got 330 pages left to read and he's going to listen we write this let me write the unit's down to pages and he's reading at a rate of 55 pages per hour pages per hour this is the same thing this is going to be equal to 330 pages times 1 over 55 hours per page I'm dividing by something the same thing as multiplying by its reciprocal so 55 pages per hours if you divide by that that's the same thing as multiplying by 155th of an hour per page is one way to think about it and so what do you get the pages cancel out pages divided by pages and you have 330 divided by 55 hours 330 divided by 55 hours and what's that going to be let's see 30 divided by 5 is 6 300 divided by 50 is 6 so this is going to be equal to 6 hours now we have to be very careful you might want to write 6 hours here but this is 6 hours after this point 6 hours after that point so in total it's going to take him 10 hours at 4 hours you have 330 pages left to read and then there's 6 more hours to finish those 330 pages so it's going to take a total of 10 hours and we could also see if this makes sense if he reads 55 if he reads 550 pages in 10 hours if he reads 550 pages in 10 hours in 10 hours what's his rate of reading well v 550 divided by 10 that's going to be the same thing as 55 pages per hour which is consistent with what we just read that he reads at 55 pages per hour now another way that you could have tackled this you could say hey this is going to be a linear equation is what if you're well versed with your linear equations you could say look he's reading at a constant rate and if we're talking about let's say let me just set up a function here we could say left to read as a function of time is going to be equal to he's reading at a rate of 55 pages per hour so the amount he's left to read is going to go down by 55 pages per hour so it's going to go down 55 55 pages every hour so you could write it like that plus so whatever the initial number of pages were plus whatever the initial number of pages were and then you could use this information right over here to solve for B we know that when time is equal to 4 hours we know that when let me do it in that color we know that when time is equal to 4 hours that the page is left to read there are 300 30 pages left to read or 330 and so you get 330 is equal to let's see negative 55 times 4 is negative 220 negative 220 plus negative 220 plus B plus B and then if you add 220 to both sides 220 add 220 to both sides you're going to be left with on this side you're going to get 550 is equal to these cancel out they add up to zero is equal to B and so the function that describes how much he has left to read the amount he has left to read as a function of time is what he's reading 55 pages per hour so the amount he has left to he's going to go down every hour that's why we have that negative there so negative 5 the negative 55 pages per hour times the number of hours plus 550 so if someone said well how long is the book well that's that's going to be L of 0 that's going to be how many pages he has left to read at time 0 and we see that L of 0 is going to be 550 when T is equal to 0 let me write this down L of 0 is going to be equal to 550 when T is equal to 0 this term goes away and you just have 550 pages and just like that we could have answered this question and then you could say well look if the book is 550 pages and if he's reading it 55 pages per hour how many hours is going to take well 550 pages divided by 55 pages per hour is going to get you 10 hours now we could have done that here actually now that I think about it I'm saying 330 pages divided by 55 pages per hour and say hey that's going to take us six more hours from this point we already knew this and we could have said look look forget if we have 550 pages to read and we're reading at 55 pages per hour it's going to take us 10 hours so you see there's a ton of different ways to solve these problems and they're all really at the end of the day representing the same thing that's what's neat about mathematics they're just ways of of representing weight thing how to think about things and at the end of the day we're really just thinking about the same problem just representing it with with different ways and different symbols