| You drive down the road at 55 miles per hour for 2 hours. | ||
| How far did you go? | ||
| Some people can do this one in their heads. | ||
| The answer is 110 miles. | ||
| The actual formula for doing problems like this one is ... | ||
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| So for the problem we started with ... | ||
| Speed you were going = 55 mph | ||
| Time you were traveling = 2 hours | ||
| Distance traveled = ? (what we need to find out) | ||
| So: | ||
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Distance Traveled = 55 × 2 = 110 |
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| Remember the page on percents? | ||
| We said that percent was the combination of per | ||
| (meaning divided by) | ||
| and cent (meaning 100). | ||
| Here, when we say 55 miles per hour | ||
| the per means the same thing, divided by. | ||
| So 55 miles per hour actually means | ||
| "55 miles divided by one hour." | ||
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| In the problem we multiplied this by 2 hours. | ||
| What we were actually doing was: | ||
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| Hey, this one looks like one of those | ||
| unit conversion things! | ||
| Well, that's part of what's going on here. | ||
| We cancel the hours in the numerator | ||
| with the hours in the denominator. | ||
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| We don't need to write in a denominator of 1 | ||
| so we can lose it ... | ||
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| The moral of he story is this. | ||
| If the problem said | ||
| "You were going 55 miles per hour for 20 minutes," | ||
| you would have to convert the 20 minutes | ||
| to whatever that is in hours before you could do the problem. | ||
| Here's a picky point. | ||
| Generally in these problems, they don't use the word speed. | ||
| Instead, they use the word rate. | ||
| Just remember that rate means speed. | ||
| It's a thesaurus thing I guess. | ||
| Example: | ||
| You drive down the highway at 65 miles per hour for 20 minutes. | ||
| How far IN FEET did you go? | ||
| OK, OK, they want the answer in feet. | ||
| That's no big deal. | ||
| Somewhere along the line in the problem, | ||
| we need to convert our measures to feet. | ||
| We could work the whole problem in miles if we want | ||
| and then convert to feet at the end. | ||
| We could also change 65 miles per hour | ||
| to whatever that is in feet per hour to begin with | ||
| and then work the problem with that. | ||
| Do whatever is easier for you. | ||
| It doesn't matter. | ||
| If you do the math right, you'll get the right answer. | ||
| Distance = (What you want to find out) | ||
| Rate (means speed) = 65 miles per hour | ||
| Time = 20 minutes | ||
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| So we've got hours and minutes in the same problem. | ||
| One of them needs to be changed | ||
| to be the same units as the other. | ||
| The easiest way is to change 20 minutes | ||
| to some amount of hours. | ||
| Remember how we do it? | ||
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| So now we can do the problem ... | ||
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| The little line over the 66 means that this number | ||
| (the 6's) repeat forever. | ||
| So now we just need to change miles to feet. | ||
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| Let's do another one ... | ||
| Example: | ||
| We drive for 4 hours and go 190 miles. | ||
| What was our average speed? | ||
| We might not have been going the same speed | ||
| for the whole 4 hours, | ||
| so we want the average speed for this time ... | ||
|
Distance Traveled = Rate × Time |
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| Distance = 190 miles | ||
| Rate = (We don't know) | ||
| Time = 4 hours | ||
| So ... | ||
|
190 miles = Rate × 4 hours |
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| Let's use R to stand for rate and solve the problem for it. | ||
| The math works just the same with an R | ||
| as it does with an X ... | ||
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190 miles = Rate x 4 hours |
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| Divide both sides by 4 hours and simplify ... | ||
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| The rate (speed) was 47.5 miles per hour. | ||
| OK, one last one ... | ||
| Example: | ||
| Two trains start at the same point. | ||
| One train travels east at 40 miles per hour for at least 4 hours.. | ||
| The other train travels west at an unknown speed. | ||
| 4 hours later, the trains are 360 miles apart. | ||
| What was the average speed of the second train. | ||
| OK, now we have 2 moving objects. | ||
| The distance we are given is the total distance they went together | ||
| (Since they were traveling in opposite directions). | ||
| We have: | ||
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360 = Distance traveled by train 1 + Distance traveled by train 2 |
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| We know that: | ||
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Distance Traveled = Rate × Time |
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| So; | ||
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360 = Rate of train 1 × Time + Rate of train 2 × Time |
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| And we know: | ||
| Rate of train 1 = 40 miles per hour | ||
| Rate of train 2 = we don't know; call it R | ||
| Time (for both) = 4 hours | ||
| So we have: | ||
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| The rate of the second train is 50 miles per hour. | ||
|
copyright 2005 Bruce Kirkpatrick |
