# Solving Distance Problems 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.

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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.

I'm assuming that they want our average speed in hours.

Average speed for the entire trip is going to be equal to the total distance, which is 12 miles, divided by her total time. So her average speed is 12 over 1, which is just 12 miles per hour.

So now we're ready to calculate her average speed for the entire trip. So what you really have to do is just think in terms of go back to your basics-- total distance, total time. This first sentence right over here gives us half of the total distance, the time to the store.

It's the same as the distance to the gift store.

Actually, let me write that in the same green color since I'm writing all the times in green color. Now, we know that the distance to the gift store and the distance back from the gift store is the same. 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. That's going to be 24 over 4 which is equal to-- did I get it? So that's why I just said that the total distance is just going to be two times the distance to the gift store. So it's going to take her-- actually, she went there much slower than she came back. So it's going to be 3/4 hours is the time times an average speed of 8 miles per hour. But that wouldn't have been right, because she's traveling those for different amounts of time. And then we get total distance divided by total time-- 12 miles per hour. And then we can figure out the distance from the store and using that and the speed to figure out her time back.d/30 (100-d)/40 = 3 Solving which gives d = 60, which is the distance travelled by train.100-60 = 40 miles is the distance travelled by bus. A plane covered a distance of 630 miles in 6 hours. part of journey to be ‘d’, the distance covered in the second half becomes ‘630-d’.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. 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 45 minutes in hours, so it's 45 minutes out of 60 minutes per hour. And what's the time coming back from 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. We're going to have 6 over 24 is the same thing as 1/4. And then miles divided by miles per hour is the same thing as miles times hours per mile.

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