Word problems with multiple units
Jamir is training for a race and is running laps around a field. If the distance around the field is 300 yards, how many complete laps would he need to do to run at least 2 miles? So they tell us how far one lap is, it's 300 yards, but we need to figure out how many laps to go 2 miles. So a good starting point would be to get everything into the same units. We have distance here in terms of miles, we have it here in terms of yards. So let's just get everything into yards. So he needs to run 2 miles. How do we convert that to yards? Well, I don't have it memorized how many yards there are per mile, but I do have it memorized how many feet there are per mile. And it's a good thing to have in the back of your brain someplace, that in general you have 5,280 feet per mile. It's a good number to know. 5,280 feet per mile. So if we want to convert, we can first convert the miles to feet, and then we know that there are 3 feet per yard, and then we'll have 2 miles in terms of yards. So 2 miles, if we want it converted to feet, we want miles in the denominator and we want feet in the numerator. And the reason why I say that is so that this miles will cancel out with that miles, and we'll just have feet there. And I just wrote down, there's 5,280 feet per mile, or you say 5,280 feet for every 1 mile. You can write it either way, but let's just write it like that. And then we can multiply. So this is going to give us what? If we just multiply the numbers 2 times 5,280. So what is that going to be? Maybe I should get a calculator out. Or we could do that in our head. Let's think of it this way: 2 times 80 is 160. 2 times 200 is 400. So it's going to be 400 plus 160 is going to be 560. And then 2 times 5,000 thousand is 10,000. So it's 10,560. And then the miles cancel out, and we are just left with feet. And let me actually multiply it out. I did it in my head that time, but that's not always useful. Let me verify for you that 5,280 times 2 is indeed 10,560. So 2 times 0 is 0. 2 times 8 is 16. Carry the 1. 2 times 2 is 4, plus 1 is 5. 2 times 5 is 10. 10,560. So he needs to run 10,560 feet. Now, we want this in terms of yards. So 10,560 feet. Let's convert this to yards. Well, we want it in yards. So we want yards in the numerator, and we want feet in the denominator, so that the feet cancel out with that feet right there. And we know that there are 3 feet for every 1 yard. Or another way to read this is that you have 1/3 of a yard for every foot. And now we can multiply. And it makes sense. If you're going from feet to yards, the number should get smaller because yards is a bigger unit. You need fewer yards to go the same distance as a certain number of feet. So it makes sense that we're dividing. Same thing: 2 miles is a ton of feet, so it made sense that we were multiplying by a large number. Here it makes sense that we're dividing. So let's do this. So this becomes 10,560 times 1 divided by 3. So it's 10,560/3. That's that and that part. And then the feet cancel out, and we are just left with yards. So 2 miles is 10,560 divided by 3. And let's figure out what that is. So 3 goes into 10,560. It doesn't go into 1. It goes into 10 three times. 3 times 3 is 9. And we subtract. We get 1. Bring down this 5. It becomes a 15. 3 goes into 15 five times. 5 times 3 is 15. We have no remainder, or 0. You bring down the 6. 3 goes into 6 two times. Let me scroll down a little bit. 2 times 3 is 6. Subtract. No remainder. Bring down this last 0. 3 goes into 0 zero times. 0 times 3 is 0. And we have no remainder. So 2 miles is the equivalent to 3,520 yards. That's the total distance he has to travel. That's the equivalent of 2 miles. Now we want to figure out how many laps there are. We want this in terms of laps, not in terms of yards. So we want the yards to cancel out. And we want laps in the numerator, right? Because when you multiply, the yards will cancel out, and we'll just be left with laps. Now, how many laps are there per yard or yards per lap? Well, they say the distance around the field is 300 yards. So we have 300 yards for every 1 lap. So now, multiply this right here. The yards will cancel out, and we will get 3,520. Let me do that in a different color. We will get 3,520, that right there, times 1/300. When you multiply it times 1, it just becomes 3,520 divided by 300. And in terms of the units, the yards canceled out. We're just left with the laps. So this is how many laps he needs to run. So 3,520 divided by 300. Well, we can eyeball this right here. What is 11 times 300? Let's just approximate this right here. So if we did 11 times 300, what is that going to be equal to? Well, 11 times 3 is 33, and then we have two zeroes here. So this will be 3,300. So it's a little bit smaller than that. If we have 12 times 300, what is that going to be? 12 times 3 is 36, and then we have these two zeroes, so it's equal to 3,600. So this is going to be 11 point something. It's larger than 11, right? 3,520 is larger than 3,300. So when you divide by 300 you're going to get something larger than 11. But this number right here is smaller than 3,600 so when you divide it by 300, you're going to get something a little bit smaller than 12. So the exact number of laps is going to be a little bit lower than 12 laps. So 2 miles is a little bit lower than 12 laps. But let's make sure we're answering their question. How many complete laps would he need to do to run at least 2 miles? So they're telling us that, look, this might be, 11 point something, something, something laps. That would be the exact number of laps to run 2 miles. But they say how many complete laps does he have to run? 11 complete laps would not be enough. He would have to run 12. So our answer here is 12 complete laps. That complete tells us that they want a whole number of laps. We can't just divide this. If we divide this, we're going to get some 11 point something, something. You can do with the calculator or do it by hand if you're interested. But we have to do at least 12 because that's the smallest whole number of laps that will get us to at least this distance right here, or this number of laps, or the equivalent of 2 miles.