Comparing with multiplication and addition: money

So we're told that Minli has $5.00 more than James. Minli has $33.00. How much does James have? So let's use some letters to represent the amount of money Minli has and the amount of money James has. So let's say that m represents the amount of money that Minli has. And j represents the amount that James has. So we're told that Minli has $5.00 more than James. So Minli's money, or the amount of money Minli has, is going to be equal to the amount of money James has plus 5. Start with the amount James has. Add 5, you have Minli's. Minli has $5.00 more than James. Now they tell us, Minli has $33.00. So let me write this down. So we know that m is equal to-- so this the sentence right here tells us, this right here tells us, that m is equal to $33.00. So instead of an m here, we could say, well $33.00 is equal to the amount of money James has plus 5. And so now we just need to think about how much money does James have. And I encourage you to pause the video and think about it on your own. Well, one way to visualize this is maybe on a number line. So let me draw a number line right over here. And let's say this is 0. This is the amount that Minli has right up here. So Minli has $33.00. That represents that point right over here. And j is the amount the James has. So let's say that this is j right over here. And we know that if we add 5 to j-- if you take j, add 5, you get to 33. Well how can I go to 33? How can I start with m or how can I start with the amount Minli has and then end up with the amount James has? Well, I could just go the other way around. I could start with m, and I could subtract 5. So we could say that 33, which is m, the amount of money Minli has, minus 5, is equal to the amount of money that James has. So how much money does James have? Well, 33 minus 5 is going to be 28. So we can say that James has, 28 is equal to j, or James has $28.00. So this right over here is 28. And you see that. 28 plus 5 is 33. 33 is 5 more than 28. So this all works out. Now let's think about this next question. Jessica's house is 5 times as far from school as Paulette's house. Jessica's house is 15 miles from school. How far is Paulette's house from the school? So just like we did, let's just look at each of these sentences and see what they're telling us. And I encourage you to define use of letters to represent the distance Jessica's house from school and Paulette's house from school. And try to figure this out on your own. Pause the video right now, and try to figure it out on your own. So you could imagine j might be a good letter for Jessica's, the distance from Jessica's house to school. And let's say that p we'll use for Paulette's house from school. So we're told that, however far Paulette's house is from school, you take 5 times that to see how far Jessica's house is from school. So we could write that p, which is how far Paulette's house is from school, times 5, is equal to how far Jessica's house is from school. Jessica's house is 5 times as far from school as Paulette's house. p is Paulette's house's distance from school. j is Jessica's house's distance from school. They then tell us that Jessica's house is 15 miles from school. So they're essentially telling us, this sentence right over here, they're telling us that j is equal to 15. So we can rewrite this as the distance Paulette's house is from school times 5, which we know to be Jessica's distance from school, or Jessica's house's distance from school, which we now know to be 15, is going to be equal to 15. So what would p be? Some number times 5 is going to be equal to 15. Some number of miles times 5 is equal to 15 miles. Well, you might already be able to think about this in your head, but we could also visualize it. So let's draw a number line again. And if you start with that number, p-- so this is Paulette's distance from school so, p-- and you multiply it by 5. So that's times 1, times 2, times 3, times 4, times 5. So notice this is p, this distance is p right over here, this is another p right over here, this is another p right over here, that's another p right over there, and another p right over there. Or another way of thinking about it, this whole distance right over here is going to be p plus p plus p plus p plus p, or p times 5. And that's equal to how far Jessica is from school, or Jessica's house is from school. So this right over here is equal to 15, which is the same thing as j. So how would you figure it out? Well, if 5 times p is equal to 15-- if I want to figure out what p is, I could just divide 15 into 5 equal groups. So we could say that p is equal to 15 divided into 5 equal groups. If you take 15 miles and divide it into 5 equal groups, you're going to end up with p. So what is this? Well, we get p, 15 divided by 5 is 3. So Paulette lives 3 miles from school. And you see that. This is a 3, another 3, another 3, plus another 3, and another 3. 3 time 5 is 15. Jessica's house, which is 15 miles from school, is 5 times as far from school as Paulette's house, which is 3 miles from school so it all makes sense.