Math 7 4 2 Homework Help Morgan

Hey, this is mass 7 unit 4 lesson two looking at ratios and rates with fractions. Today's lessons will be little more clear than the last one. Sorry about the last one. just had lots of things going on with it. But today we today we kind of bring it just little more compact. All right. So first of all, we have number talk division. find each quotient mentally. When we're dividing something, remember what's happening is we're multiplying by the reciprocal. And so look at these, the easiest way to look at that would be to go down like that and think 5 * 3 is 15. Again, looking at that, what do you see along the diagonal happening right there? We would say 2 * 3 is 6. Over here, when multiply that up there, just get three. That can stay one. So end up with 3 over 2. This one though, we'd have to go 2 * 2 is 4 + 1 is five. So actually have 5 over two is what this one really ends up being. And then end up with 15 over two for that one there. Anyways, just fun little quick little number talk there looking at division and finding quotients mentally. So moving on to today's lesson. It says that train is traveling at constant speed and goes 7.5 km in 6 minutes. At that rate, how far does the train go in one minute? All right. So what we're talking about here is this. We're saying that you have train that is traveling certain distance. Okay? And it does that distance in certain amount of time. In our case here, we know that it's going 7.5 km and it's doing 7.5 km in 6 minutes. That is the rate that that train is traveling. 7.5 km in 6 minutes. Okay. So that's what we know so far. All right. So in terms of what this train is doing, it's doing 7.5. that's it. Now the question it wants to know is how far is it going in 1 minute? Okay, that's the question it wants to know is how far is it going to go in 1 minute? Okay, and that's what we have to figure out. So what we want to think about is again we've talked before about this being our and this being our And our constant proportionality is our overx. So 6 over 7.5. Okay. could think of this also as being 6 / 7 and 1/2 because that's what 7.5 is. All right. And so this becomes 2 * 7 is 14 + 1 is 15. So it becomes 6 / 15 / 2. can write that as 6 * the reciprocal 2 over 15. And when do that there, can reduce. Three goes into that two times and three goes into that five times. So end up with value of four fths. That's my constant proportionality. Now that means that if take number here and multiply it by 4 fths, end up with what's over there. But that's going that direction. When want to go, if you recall, the opposite direction, I'm going to multiply by the reciprocal. So instead of 5/4s, we multiply by 4 fths, we multiply by 5/4s, the reciprocal. So 1 * 5/4s, it's going to be 5/4s. So in terms of how far does train go in 1 minute, it's going to go 54 km in 1 minute. So that's what we would do to solve that problem. Like so. How far does train go in 100 minutes? Well, again, we have 100 minutes here. Okay. And if multiply by my reciprocal going this way, 5 over 4, we end up with 500 over 4. And 500 over 4 = 125. All right. So, that's one way of looking at it there. If by chance you decided, you know what, wanted to do time. I'm just going to just not worry about this right now. want to do time over here and want to do distance over here. Would that change what you came up with? If my time was six and my distance was 7.5, what would change? Well, here's what would change. would still have constant proportionality, which would now be 7.5 / 6, right? That's my new and my new So 7.5 / 6 this becomes 7 12 / 6 that becomes 15 / 2 / 6 15 /2 * 1 / 6 we reduce and we have 2 and we have 5 and so we end up with 54s is my value. Notice that it is the reciprocal of what did over here because changed my and values around. I'm also going to flip-flop my constant proportionality. So time in this case here times 5.5 over4 will get me that one which is little easier when you say what happens if have 1 minute. Well 1 minute * 54 is 54. So from reading standpoint just going left to right that flows little better than what we did up here when we went that direction there. Okay. So could say over here if it wants to do 100 could do 100 * 5 over4 and that again is 500 over4 which is the same as 125. So that seems to make much makes more sense in some way to go from time to distance. But want you to see that it just depends on how you set it up. You'll still end up with the same solutions just different way of going about it and you end up working with reciprocal in different way. Okay? So just keep that in mind that as you're working with something, you decide what your variables are going to be, your and y's, your distance or time, and you have to then work with that the whole way through the problem. You can't switch things around. So be careful. Number part three, comparing running speeds. Okay, we have Lynn, who is running 2 and 34 miles in two-fifths of an hour, and Noah over here. So here's Lynn's value, and here's Noah's value. Okay, so some questions we might be looking at here. Who ran faster? That could be question we want to know. We might want to know how far did Lynn run in one hour. Okay, we might want to find out how long would it take Lynn to run one mile at that rate. There's other questions your teacher may have for you as well. So, let's take look at Lynn first of all. So, here's Lynn. Lynn is running at rate of 2 and 34 miles. Right. Doing 2 3 34 miles per two fifths of an hour. That's our setup. 2 and 34 / 2s. So let's turn this into some improper fractions. 4 * 2 is 8 + 3 is 11. 11 / 4 / 2s. Well 2 fths becomes 5 over2. So let's move this over here. This becomes 11/4s * 5 over 2, which becomes 55 over8. And 55 over 8 as mixed number is 6 and 7/8. So how fast is Lynn going? She's doing 6 and 78 miles per hour is what Lynn is doing there. So that's Lynn. How far Lynn run in 1 hour? She's doing 6 and 78 miles in 1 hour. Now in terms of Noah, for Noah, Noah is doing 8 and 2/3 / 4/3. So again the setup is going to be 3 * is 24 + 2 is 26 over 3 / 4/3 which becomes 26 over 3 * 34s. Now the threes will cancel out. Two goes into there two times. 2 goes in there 13 times. Right? So have 13 over two, which is the same as 6 and half miles per hour. So Noah's pace is 6 and 12 miles an hour. Lynn's pace is 6 and 7/8 miles hour. Who's running faster would be Lynn. And that's because 6 and 7/8 is greater than 6 and 12. Now, how long would it take Lynn to run one mile at that rate? And that's little bit different question, right? So, we're looking at really in this case here, we're looking at the reciprocal, right? Remember the last problem, if we want to go the other way, we just flip it over. So, for Lynn, would actually take her 8 over 55 instead of 55 over 8, we take that and we're going to multiply it by the number of minutes there are in an hour. Okay? And we end up with in this case here lot. We end up with lot. We end up with 8 hours. Well, let's do it this way. We end up with 480 over 55. And that then reduces down to 8 and 81. Okay. minutes is how long it would take her to do that. So it takes her 8 minutes or about little more almost 9 minutes to run mile at that rate. Okay? And that's what she's doing. So you can take look at your partners and see what you guys come up with. But variety of things there. looking at how fast Lynn is or Noah and comparing things there. All right. Going to skip the are you ready for more and we're going to look now at the scaling the Mona Lisa. Scaling the Mona Lisa. Okay. In real life, the Mona Lisa measures 2 1/2 ft by 1 and 3/4t. company that makes office supplies wants to print scaled copy of the Mona Lisa on the cover of notebook that's 11 by 9 in. So, we're moving from feet to inches. First of all, so what size should they use for the scaled copy Mona Lisa on the notebook? Couple things you need to think about here is does it need to cover the whole notebook or you just want to put it on the notebook, right? So, that's what you think about as well. In this case here, we're probably just going to put it on the notebook. If we did something different with it, we might end up with it being stretched and twisted some way or another. So, it's kind of cool to look at. If you were to look at the little picture here we have on our website for this lesson, it gives you picture of the Mona Lisa and it says, "Well, you can kind of play little bit, right? We can move it around little bit." Oops. pushed the wrong buttons. No, back up. Back up. Back up. Stop it. Stop it. Stop it. and see what I'm doing. need to go back on my arrow. So, what we can do is we can actually move this around. It's important to know that when I'm going to put this on notebook over here, need to make sure don't squish it, right? If make it too narrow, change what it looks like. If push it down this way, then flatten it out. That's not going to work. Even though can get it on part of that, need to make sure that whatever I'm doing to it, keep it about the right size so that keep its dimensions the right way. Now, here I'm putting it on there, but I'm not sure if that's proportional or not. It just looks like it's going to fit. But look when look at my my bars right there, notice that they're not in the same spot, are they? To put them in the same spot, the exact same spot or pretty close to it. end up not covering the whole shape there. Okay. So, it's interesting how this looks. It just shows you that when put it in the space, I'm not going to be able to cover up the whole cover with the Mona Lisa. Doesn't work out that way. But when we work on reducing that to fit or making scale copy, want to keep the dimensions accurate so end up with still the Mona Lisa and not the, you know, really squished one. We don't want that. So let's take look then at what you have. So what size should they use for the scaled copy of the Mona Lisa on notebook? All right. First of all, let's take our our information. We know we have feet and we're going to go from feet and convert feet into inches. All right. So to do feet and inches, let's take the height here 2 and half, which is 2 * 2 is 4 + 1 is 5. So 5 over 2. and we're going to take it from feet and put it in inches. We're going to multiply by 12 to do that. Okay? Why? Because there are 12 in in 1 foot and this is feet. So the feet will cancel out. So that becomes 12id 2 is 6. 6 * 5 is 30 in. So we know that part and the 1 and 34 is 4 + 3 is 7. So we do 7/4 * 12. Again, same idea. 1 and three. Reduce. 7 * 3 is 21 in. So, know I'm working with 30 in by 21 in at the actual dimensions of the Mona Lisa. Okay. So, 30 and 21. So, that's my first bit of information I'm working with there. got to get it onto book that's 11, right? 11 by 9 is what I'm aiming for there. 11 by 9. So in estimating here, would say that the difference between 11 and 30 is about maybe we might say maybe it's about third, right? And why do say third? Well, 11 it's like about 10 and 30. I'm just just estimating here. So 10 out of 30 is about 1/3. So it's something like about third of what the original was there. So, if had 30-in height and multiplied that by third, would be able to use height of about 10. And if did original of 21 * about a/3, 21 * third is about 7. Okay. So, might have dimensions of something like 10x7 for what the de Mona Lisa would look like there. So would it work with 11 by9? Sure it would. would just have some space to spare. So if my if my book cover is 11 by9, my Mona Lisa is going to be 10. So get most of that and not quite fill in the real thing there. So I'm going to have some extra space around there. I'm not filling in the whole thing. Again, going back to our picture before, when took look at the Mona Lisa there, we pop it into place here. When slide this and drop it into that corner, something like that. If reduce these things, can't do the same time, but if shrink that down to something like that. Let's see what that dimension is going to be. We'll get there in second. All right, we're saying 10 by 7. Something like that. Let's see. To put them the same. Oops. Are those lined up? Not quite. So, let's go little bit more this way. It's hard to do with my finger. Looks pretty close there. Something like that. Those look pretty lined up. We could check with our centimeter and see what we have. That says it's about want perimeter. no perimeter. Back up. Back up. Forward's good. All right. Let's get our centimeter thing here. And let's find point. and point. It's going to me perimeter still 7.7 there. And then over here it's telling me that it's 11. So that's 11. We wanted to go with 10. Anyway, so you can play with this some more. Again, not this is just an estimate here. Anyways, so get the idea. We could make fit on 10x7. If it's 10 7, we can make fit there. No problem at all. So what's the scale factor? What is the scale factor from the real paying to its copy on the notebook cover? So in this case to go from the copy to the notebook cover the notebook cover is 11 and the original is 30. So our scale factor would be 11 to 30 would be what the real scale factor should be if we want to make it work perfectly. There is the idea. there you go. So let's take look the lesson summary. The lesson summary just kind of reviews how to do some conversion stuff to convert foot to inch and how we move some things around little bit there. Let's take look at tonight's homework. Okay. Tonight's homework says cyclist rode 3.7 mile 75 miles in.3 hours. How fast was she going in miles per hour? Okay. So, here we have hours and here we have miles. Make ourselves little chart here. and Okay. And we know that it went.3 in.3 hours she went 3.75 miles. Now, the reason set it up this way is that we're talking about miles per hour. Notice here miles per hour. So, that's why put it in this way. So, I'd have my over my So in terms of miles per hour, we would say that it's 3.75 /.3. And we do that. could look as decimal, which is one way to look about it. If you're using calculator, that could work there. But in this lesson, what they're looking at as well is to say, well, 3.75 is the same as 3 and 3/4s. And this 3/10 is the same as we're going to divide by 3/10 3/10 there. So that becomes 12 + 3 is 15 over 4 times reciprocal 10 over 3. Reduce the 3 and the 15 becomes 1 and 5. The four becomes 2 and 5. So have 25 over 2 which is equal to 12.5 miles hour. Right? So lots of ways of looking at that's the fractional way. If just did the decimal way like this with calculator would do 3.75 /.3 end up with 12.5. So it's little bit faster sometimes to work with the decimals on calculator but that's how you could convert to fraction and then solve it there. So miles hour is 12.5 miles hour. At that rate, how long would it take her to go 4.5 miles. So what are we saying is here in 1 hour we can do 12.5 miles. And the question then is if have 4.5 on my chart there, how many hours would it take? So we're going the other way around, right? So instead of multiplying by 12.5 because here it was multiply by 12.5 to go this way we will divide by and this should write this way divide by 12.5 really to say well how's that going to work out so we do 4 12 / 12.5 which is 12 1/2 so 4 1/2 / 12 1/2 this becomes 9 / 2 / 12 * 2 is 24 + 1 is 25 / 2. And let's do this right. So we do 9 /2 multiply by the reciprocal 2 over 25. The twos will cancel out and we're left with 9 over 25. All right. And that's going to be how many? Well, that's how many 95ths of an hour. in terms of how many minutes that's going to be. We're going to multiply that by 60 because we know it's going to take her 95ths of an hour to get there. Okay, which that's about 36 hours. But if want to know how many minutes that would be, multiply by 60 to get that into minutes. Okay. So that would go into 25 and 60 you can reduce into 5 and 12. So then you have 108 out of five which can become 21 minutes and 35ths of minute. So 21 and 35ths of minute. So both answers are okay. You could say it's 36 hours or 21 and 35ths minutes for how long it would take to go 4.5 miles. All right, let's take look at the next one. Number two, it's helpful to make these little tea charts here as we go through. So, let's keep doing that. recipe recipe calls for sparkling grape juice, one and half quarts and sparkling water, 3/4 sorry. So, to make sparkling grape juice, we have half quart of water and three quarts of grape juice. So, we have water and we have grape juice. That's what we have so far. Okay. So what we know is that it takes 1 and 12 quarts of water for 34 of quart of grape juice. That's what we know so far. In terms of our our constant proportionality, we have 3/4 / this number, the value / 1 and 1/2. 1 and 1/2 is the same as 3 over2. So we're multiplying though by the reciprocal. So that becomes 34 * 2/3. The ones are going to go away and we just have 24 which becomes half. So my value is 1/2. That's my value. So to go from here to here, it's times half, right? Times half. No problem. That also means So that means sorry. so what do know by that? That means that if want if have one thing of water, would have one quart of water is equal to half quart of grape juice. also can go the other way and know that if have one quart of grape juice multiplying by the reciprocal, 1 * 2 is two. So also know these values, right? So when have one quart of water, can have half quart of grape juice. If have two quarts of water, have one quart of grape juice there. So, what they're asking me though to find out though is what do do with nine quarts of grape juice? So, we're going to put nine right there, right? And this tom made it go all the way down for space. So, if have nine, I'm going to multiply by the reciprocal of the value. value here was half. So, multiply by two. So, 9 * 2 is 18. So, we're going to have 18 quarts of water. If have nine quarts of grape juice. Now, how much grape juice would need to mix with 154 quarts of sparkling water? So, we do 154s times our value, half. All right. And that becomes 15 over 8. 15 over 8, which is the same as 1 and 7/8 quarts for grape juice. Okay, so that's one answer. And here's our other answer there. How much of each ingredient would you need to make 100 quarts of punch? All right, let's take look at this value right here. This is an important one because going to use two waters and one grape juice for the whole batch, right? To make my stuff. That's what I'm going to use. Two waters and one grape juice. One quarter grape juice to make the stuff there. Now, what this means though is these are quartz, so it's 2:1 ratio. Okay, so let's change this to an and an Okay, don't know how much of each I'm going to have, but know I'm going to have two times what have for two times the water that have for the grape juice. That's what know. Now, altogether, that's three of my X's all together. And want to make total of 100 quarts. So, I'll set that equal to 100. If divide both sides by 3, that gives me my value. All right, = 100 over3. So, what does that mean for us here? Well, what it means is if I'm going to make 100 quarts of punch. Okay, let's see how much of the of the grape juice I'm going to make. I'm going to use this value for I'm gonna put it in here for my grape juice or sorry my water. Yep. So water is going to be 2 * 100 over3 which is 2003 quartz. And the grape juice is going to be 1 * 100 over3 which is going to be 100 over3. So I'm going to use 203 quartz and 103 quarts in order to make the recipe the right way. All right. So there you go. Next one. Draw scaled copy of the circle using scale factor of two. So we're going to be multiplying things by two right here. Let's see what that's picture looks like. All right. So here's our picture. First of all, I'd take the picture here and I'm going to measure that across. And see that it is about looks like from side to side I'm about 1.5 in is my diameter. If I'm going to do scale factor of 2, then I'll do 1.5 * 2, which is 3 to make my new diameter. So if three is my new diameter, then the halfway point will actually be at 1.5. And can draw line here to three for new diameter. could draw another one as well just to have couple more points I'm working with there. All right. So there to there. And then could play connect the dots and draw circle looking something like that. Not very good circle, know, but gets you the idea. So that's my new diameter is three there. The next question says, okay, let's look at 3B. It wants to talk about the circumference. How's the circumference of the scaled? So scale compared to the original. All right. So circumference is pi * the diameter. Right? Now our diameter went from 1.5 to 3. So what was happening there? We were going to be multiplying by two. So it's two times larger. The circumference of the new circle is going to be two times larger than the original circle. This third question here wants to know about the area. What's the difference in the area between the two things there? Okay, so what happens with the area remember is that area is going to be the radius squared. So what happens though for the area and we look at what happens is you take scale factor and you square the scale factor. In our case our scale factor was 2. So 2^2 is 4 and so the area is going to be 4 * larger than the original area because it's squared value for the radius there. All right let's look at the next one. Someone bought 1 and 3/4 lbs of ham for 1450. Someone bought 2 and a2 pounds of turkey for that and roast beef for that. All right. Which meat is the least expensive and which is the most expensive per pound? All right. So, we have some math problems to work out here. Okay. What we're doing is is the the cost per pound. That's going to be written like this. This is going to be the dollar amount divided by the pounds. So if you're thinking about it like this, we have ham first of all, which is $14.50 divided by 1 and 34s. Okay, that's our ham. We have our turkey. Turkey is going to be $26.25 divided by 2 and half. And then we have our roast beef, which is going to be $5.50 divided by 38 of pound. And so these are the three problems we're going to have to do to figure out what's the least expensive and the most expensive. All right, so how can we do this here? Let's rewrite 1 and 3/4s as mixed or an improper fraction. So 1 and 34s becomes 4 + 3 is is sorry 4 + 3 is 7/8. Okay. So 78 here over here 2 and 1/2 becomes 5 over 2 and that's already there at 38 there. Okay. So we're looking at so far right 4 5 6 7 8. Yep. So now we're going to multiply these by the reciprocal. So, we're going to flip them over and go 8 over 7, 2 over 5, and 8 over3. So, 1450 * 8 / 7. Okay, get the calculator out here. 14.5 * 8 = that. Divide that by 7 and we have $16. Is that going to be right there? Sure. what' do? Sorry. There's something wrong there. Let's fix that there. That's four 567. So, it's 7/4s. Sorry about that. This just becomes 47th. knew something was funky in my math there. So, we do 14.5 * 4. 4 equals that amount. Divide that by 7 and we have 8.28. So about $828 pound. Over here we do 26.25 * 2 / 5 and we end up with $10.50 pound. And finally over here 550 * 8 / 3 is about $1466 pound. So, our least expensive is going to be the ham. Okay? And our most expensive is going to be the roast beef. And that's how we solve that there. And then the last one here today, I'm going to have our kiddos skip this one here, but we can do the ones, we're going to go into the ones with just the miles. I'm not going to worry about the centimeter part there. So, I'm not going to worry about these two. Okay? I'm not worry about this one and this one. Wait, which one am wor about? Sorry. Yeah, Okay, we have some ones that deal with centimeters. So, to work on those, you have to convert some stuff over. But let's take look here. You have page that is 5 in, okay, by 8 in, something like this. and you have Kansas that is 210 miles 210 miles by 410 miles. So what would work for scale of that map? Can make 1 in equal to 1 mile? In this case here, that wouldn't work because that only be 5 miles. Just not possible. Can make an inch equal to 10 miles? 10 20 30 40 50. need to get to 10. Not quite enough. Could make foot equal to mile? Well, only have 5 in. So don't even have foot. That's not going to work for that one. And can make an inch equal to five to 100 miles and make it work? Sure. One, two, three. can make that fit right there. So, wouldn't take the whole page. That would definitely work. So, is great answer there. Okay. For the other ones here, you have to be able to convert some stuff around. Not going to mess that with you right now today because just time stuff. but there's only one answer of those three. So, I'll let you figure that one out. See what your teacher says. All right. Have great day.