| If you wanted to find out how many feet were in two yards you might say | ||
| "Well let's see, there's 3 feet in one yard so there must be 6 feet in 2 yards." | ||
| That works just fine. | ||
| But what if you wanted to find out how many inches there are in 2,500 miles. | ||
| Or find out how many feet per second there are in 55 miles per hour. | ||
| Then we need a better plan. | ||
| Good news! We have one. | ||
| Here it is: | ||
| Take another look at that | ||
| "How many feet are in 2 yards?" problem. | ||
| We have 2 yards, and we want to have our answer in feet. | ||
| To get there, we need to do something very, very tricky. | ||
| We need to multiply ... | ||
| by 1. | ||
| OK, OK, there really IS a trick. | ||
| The trick is the way we write the number 1 that we use to multiply. | ||
| Any time we have a fraction | ||
| with the same amount on the top and the bottom | ||
| we have a fraction equal to 1 (except when that amount is zero). | ||
| Anyway, | ||
| if we have a fraction with 1 yard on the top and 1 yard on the bottom, | ||
| we have a fraction equal to 1: | ||
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| We also know that 1 yard is equal to 3 feet. | ||
| So instead of 1 yard, | ||
| we can write 3 feet for either the top part | ||
| or the bottom part of our fraction | ||
| and we haven't changed anything. | ||
| Like this: | ||
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THIS IS OUR BIG NEW TRICK ON THIS PAGE! |
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| The trick is the way we write the fraction that is equal to 1. | ||
| We write it with things that LOOK different on the top and the bottom, | ||
| but are actually worth the same amount. | ||
| Now we have our new special name for 1. | ||
| We are ready to multiply the 2 yards: | ||
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| To make things even out, we can put a denominator of 1 under the 2 Yards. | ||
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| OK, We did it, but look! | ||
| What kind of a thing is "6 Yards x Feet" ??? | ||
| Who Knows! | ||
| And if we had to leave it looking like that | ||
| we wouldn't have anything too great. | ||
| But look! We have the word yards in the top | ||
| and the word yard (close enough) in the bottom part. | ||
| We can cancel them out. | ||
| You're kidding! | ||
| Nope! Works every time. | ||
| The only thing is that everything on top and everything on the bottom | ||
| must be multiplied. No addition or subtraction. | ||
| So ... | ||
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| That means: | ||
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2 Yards = 6 Feet |
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| Hey, we KNEW that 6 Feet was the answer all along. | ||
| So this is no shock. | ||
| But let's go over exactly how we come up with the funny name for 1. | ||
| The fraction gets units that are what we have to begin with | ||
| and what we want to end up with. | ||
| We had yards and wanted to get to feet. | ||
| We know we want feet for 1 number and yards for the other. | ||
| Which gets which? | ||
| Do we want: | ||
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| Here's the deal. | ||
| Put the units you want on top and the units you have on the bottom. | ||
| The units you have go on the bottom so you can cancel them! | ||
| We had yards to begin with, so the one we want is: | ||
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| Try another one ... | ||
| Example: | ||
| Suppose we're really bored and want to find out | ||
| how many feet there are in 2500 miles. | ||
| You look up (or maybe you know) that there are 5,280 feet in a mile. | ||
| That is: | ||
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5,280 Feet = 1 Mile |
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| Since these 2 things are worth the same amount, | ||
| we can use them in a fraction as our names for 1. | ||
| We have miles and want feet, | ||
| so miles goes on the bottom and feet goes on the top: | ||
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| Now we multiply 2500 miles by this fraction that's equal to 1. | ||
| So everybody has a denominator, we put a 1 under the 2500 miles | ||
| (a calculator really helps here): | ||
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| Since we have miles on the top and mile on the bottom, | ||
| and everything is multiplied, | ||
| we can get rid of them. | ||
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| Hey that's a lot of feet! | ||
| OK, One more and we're done. | ||
| Example: | ||
| Tennessee Ernie Ford asks you to change 16 tons to ounces. | ||
| If you look in a reference book, you will find that: | ||
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1 ton = 2000 pounds |
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| You will also find that | ||
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1 pound = 16 ounces |
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| But you probably won't find how many ounces are in a ton. | ||
| So here's the plan. | ||
| First we change 16 tons to some number of pounds. | ||
| Then we change that number of pounds to ounces. | ||
| Ready? | ||
| OK, we have 16 tons and want pounds. | ||
| That means pounds goes on the top and ounces goes on the bottom. | ||
| 2,000 Pounds = 1 Ton. | ||
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| We have Tons on the top and Ton on the bottom. | ||
| Everything is multiplied. | ||
| So we can cancel tons of stuff. (he he he ... OK, it wasn't funny) | ||
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| One part down and one to go. | ||
| Now we have pounds and want ounces. | ||
| So 16 Ounces goes on top and 1 Pound goes on the bottom. | ||
| 1 Pound = 16 Ounces. | ||
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| We have Pounds on the top and Pounds on the bottom. | ||
| Everything is multiplied. | ||
| We can cancel the Pounds. | ||
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copyright 2005 Bruce Kirkpatrick |
