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# Multiple rates word problem

## Video transcript

starting at home oh mama travelled uphill to the gift store for 45 minutes at just eight 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 total her average speed for the entire trip that's going to be equal to the total distance the total distance that she traveled over the total time over the total time 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 2 times the distance to the gift store so 2 times the distance distance to the gift store to the gift store and then what is going to be her total time well it's time to the gift store so it's time time to gift store and then plus the time coming back time coming coming back from the gift store now we know that the distance to the gift store in 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 2 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 it's going to take her actually she went 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 no point here do they say hey the gift store is this far away but they do tell us this first sentence right over here oh mama travelled uphill to the gift store for 45 minutes at just 8 miles at just 8 miles per hour so we're given a time and we are given a and we're given a speed we should be able to figure out a distance so let's just do a little bit of a side here we should be able to figure out the distance to the distance let me write it this way distance to the store distance to store will be equal to and we got to make sure we have our units right here they gave it in minutes here they have eight miles per hour so let's convert this 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 sixtieths divide both by 15 that's the same thing as 3/4 so it's going to be 3/4 hours 3/4 hours is the time times an average speed of 8 miles per hour times 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 what's going to be 24 over 4 let me just write that that's going to be 24 over 4 which is equal to to get 24 over 4 which is equal to 6 and units wise we're just left with miles so the distance to the toilet 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 where they already told that to us they already told us that it's fort it's 45 minutes now I want to put everything in hours I'm assuming that we they want our average speed an hour so I'm going to put everything in hours so the time to the gift store was 3/4 of an hour 3/4 hours 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 figure we can take this distance we can take 6 miles that's the distance to the gift store six miles divided by her speed coming back which is 24 miles per hour so divided by 24 miles per hour gives us let's see we're going to have six 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 for 1/4 of an hour to get back and that's that fits our intuition actually write that in same green color since I'm writing all the times and colors in green color so 1/4 of an hour so she went there going to the gift store was slow took her 3/4 of an hour coming back only took her 1/4 of an hour 1/4 of an hour so now we're ready to calculate her average speed for the entire trip her average speed for the entire trip is going to be equal to the total distance which is 12 miles 12 miles divided by her total time 3/4 hours plus 1/4 hour is exactly one 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 anyway 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 velocity speed to figure out her time back and then we get total tot distance divided by total time 12 miles per hour