The time t t (in hours) required to travel a fixed distance varies inversely as the speed s s (in m i / h r mi/hr ). If it takes 6 6 hours to travel a certain distance at 50 m i / h r 50 mi/hr , how long will it take to travel the same distance at 75 m i / h r 75 mi/hr ?

So in this problem, we are told that the time, t in hours, required to travel a fixed distance varies inversely as the speed, s. This is going to be in miles per hour for the speed. If it takes six hours to travel a certain distance at 50 miles per hour, how long will it take to travel the same distance at 75 miles per hour? So this is an application problem involving inverse variation. Now to solve this, let's first set up our equation. Well, we are told that the time varies inversely as the speed. So that means we are going to have our time t, and then that's going to vary inversely as the speed. Now typically this is where we would put the constant k on top. But what I can see here is in this problem, we're actually told that it's the time required to travel a fixed distance. So that constant that we're looking for is distance. So rather than putting a k, I'm going to put a d up on top just so we can recognize that that's distance we're talking about. Now keep in mind, you could solve this problem with a k as well. If you wanna stay consistent and make that a k, that's perfectly fine as well. You still should get to the right solution in this problem. But let's go ahead and see what the rest of this problem is asking for now. Well, we are now told if it takes six hours to travel a certain distance at 50 miles per hour, how long will it take to travel the same distance at 75 miles per hour? So notice we are given one instance. The time is six hours when the speed is 50 miles per hour. So let's go ahead and see what our constant distance here is going to be. Well, I can see that in this case, the time is six hours. Now the distance we don't know, but I can see the speed is 50 miles per hour. So what I need to do here is solve for that distance d. And I can do this by multiplying 50 on both sides of this equation. That's gonna get the fifties to cancel on the right side here, so I'm isolating the distance. Now what is 50 times six? Well, 50 times six is gonna be 300. And I can see that we're dealing with miles, so that's gonna be 300 miles. So that is our constant D. So going back up to my equation, I can see that our time, or t, is going to be three hundred over s, giving us our relationship, our inverse variation equation right there that relates time and speed or t and s. So now that we have this relationship, we're asked how long will it take to travel the same distance at 75 miles per hour. Well, to do this, we have 300 on top, and I can see that our speed is going to be 75. So all I need to do is figure out what 300 divided by 75 is, and that comes out to just be four. So four hours would be the answer to this problem. So this is how you can solve inverse variation problems where we have real world application. Hope you found this video helpful.