Multiple rates word problem

Starting at home, Umaima traveled uphill to the gift store for 45 minutes at just 8 miles per hour. She then traveled back home along the same path downhill at a speed of 24 miles per hour. What is her average speed for the entire trip from home to the gift store and back? So we're trying to figure out her average speed for the entire trip. That's going to be equal to the total distance that she traveled over the total time. Well, what's the total distance going to be? Well, the total distance is going to be the distance to the gift store and then the distance back from the gift store, which are the same distances. So it's really you could say two times the distance to the gift store. And then what is going to be her total time? Well, it's time to gift store plus the time coming back from the gift store. Now, we know that the distance to the gift store and the distance back from the gift store is the same. So that's why I just said that the total distance is just going to be two times the distance to the gift store. We don't know-- in fact we know we're going to have different times in terms of times to the gift store and times coming back. How do I know that? Well, she went at different speeds. So it's going to take her-- actually, she went there much slower than she came back. So it would take her longer to get there than it took her to get back. So let's see which of these we can actually-- we already know. So how do we figure out the distance to the gift store? At not point here do they say, hey, the gift store is this far away. But they do tell us, this first sentence right over here, "Umaima traveled uphill to the gift store for 45 minutes at just 8 miles per hour." So we're given a time. And we are given a speed. We should be able to figure out a distance. So let's just do a little bit of aside here. We should be able to figure out the distance to the-- actually let me write it this way. The distance to the store will be equal to-- now we've got to make sure we have our units right. Here they gave it in minutes. Here they have 8 miles per hour. So let's convert this into hours. So 45 minutes in hours, so it's 45 minutes out of 60 minutes per hour. So that's going to give us 45/60. Divide both by 15. That's the same thing as 3/4. So it's going to be 3/4 hours is the time times an average speed of 8 miles per hour. So what is the distance to the store? Well, 3/4 times 8. Or you could view it as 3/4 times 8 times 1, is going to be-- well, it's going to be 24 over 4. Let me just write that. That's going to be 24 over 4 which is equal to-- did I get it? Yeah. 24 over 4, which is equal to 6. And units-wise, we're just left with miles. So the distance to the store is 6 miles. 2 times the distance to the gift store, well, this whole thing is going to be 12 miles. 12 miles is the total distance she traveled. Now, what is the time to the gift store? Well, they already told that to us. They already told us that it's 45 minutes. Now, I want to put everything in hours. I'm assuming that they want our average speed in hours. So I'm going to put everything in hours. So the time to the gift store was 3/4 of an hour. And what's the time coming back from the gift store? Well, we know her speed. We know her speed coming back. We already know the distance from the gift store. It's the same as the distance to the gift store. So we can take this distance, we can take 6 miles, that's the distance to the gift store, 6 miles divided by her speed coming back, which is 24 miles per hour, so divided by 24 miles per hour. It gives us-- well, let's see. We're going to have 6 over 24 is the same thing as 1/4. It's going to be 1/4. And then miles divided by miles per hour is the same thing as miles times hours per mile. The miles cancel out. And you'll have 1/4 of an hour. So it takes her 1/4 of an hour to get back. And that fits our intuition. Actually, let me write that in the same green color since I'm writing all the times in green color. So 1/4 of an hour. So she went there. Going to the gift store was slow. It took her 3/4 of an hour. Coming back only took her 1/4 of an hour. So now we're ready to calculate her average speed for the entire trip. Average speed for the entire trip is going to be equal to the total distance, which is 12 miles, divided by her total time. 3/4 hours plus 1/4 hour is exactly 1 hour. So her average speed is 12 over 1, which is just 12 miles per hour. Her average speed is just 12 miles per hour. And you might have been tempted to say hey, wait. Why don't I just average 24 and 8. But that wouldn't have been right, because she's traveling those for different amounts of time. So what you really have to do is just think in terms of go back to your basics-- total distance, total time. Figure out the total distance. This first sentence right over here gives us half of the total distance, the time to the store. We just doubled that to get the time back. And then our total time we can figure out. They tell us the time to the store. And then we can figure out the distance from the store and using that and the speed to figure out her time back. And then we get total distance divided by total time-- 12 miles per hour.