How To Divide Fractions (1948)

Suppose you have a length of ribbon and you wish to measure it. You have a rule 1 ft long. You're going to see how many times the length 1 ft is contained in the length of the ribbon. It is so the ribbon is 3 ft long. When you measured, what you really did was to divide the length of the ribbon by one. donuts. Our measuring stick will be two donuts. We're going to see how many times two is contained in six. It is contained three times. So 6 / 2 = 3. Let's divide 6 by 1. 1 2 3 4 5 6 6 / 1 equals 6. The smaller our divisor is, the more times it is contained in a number. We can see that clearly if we divide six by 1/2. The answer is 12. Suppose we have the problem 17 divided by 1/3. Here are 17 discs. Our measuring stick is this 1/3 of a disc. How many 1/3s are there in 17? We know that the answer will be a large number. We can also see that it's going to take a long time to find the answer by measuring. But there's a shortcut. The easy way is to find out how many 1/3 there are in one. There are three 1/3 in one. If one contains three 1/3, then 17 contains 17 * that many 1/3. 17 * 3 equals 51. So we know that 1/3 is contained in 17 51 times. There we have the answer to our problem. 17 / 1/3 = 51. That's much easier than doing all that measuring. Let's work another problem that way. Our problem this time is 5 /ed by 1/4. First, we will find how many 1/4s there are in one. There are four. We have five ones. So 5 * 4 is our answer. It is 20. 5 / 1/4 = 20. Let's try that method once more. Our problem is 2 fths / 1/2. 1/2 is our divisor. How many 1/2s are there in 1? There are two. Then we shall multiply two by the number of ones we have. How many ones do we have? We have only two fifths of 1. So we multiply 2 by 2s. The answer is 4 fths. Our answer is a fraction because two- fifths will contain only a part of 1/2. Let's see how that might work at home. Here's a jug which contains 1/2 gallon of grape juice. Here's a large bottle which will hold two- fifths of a gallon. Our problem asks, how many times does two fifths contain 1/2? Let's try to put the half gallon of grape juice into the bottle which holds two- fifths of a gallon. Two-fifths will not contain 1/2 even once. It will contain only a fraction of it. Our arithmetic tells us that that fraction is 4 fths. And we can see that it is. Let's review what we've learned so far. We've seen that the way to divide by a fraction is first to find out how many times that fraction is contained in one. Then we multiply that number by the number of ones we have. As long as our divisor is just one part such as 1/2 or 1/3 or 17th, the denominator tells us how many times that fraction is contained in one. But suppose our fraction has two parts such as 2/3. How many times is that contained in one? Let's see. One will contain 2/3 once and there's 1/3 left over. Our divisor is 2/3. So the part which is left over is 1/2 of our divisor. The answer is therefore 1 and 1/2 or three halves. That is our divisor turned upside down. How many times is 3/4s contained in one? It is contained once and there's 1/4 left over. So the part which is left over is 1/3 of our divisor. The answer is therefore 1 and 1/3 or 4/3. Again, that's our divisor turned upside down. We've discovered an easy rule for dividing one by a fraction. If you wish to find out how many times a fraction is contained in one, just turn the fraction upside down. And there's your answer. Turning a fraction upside down is called inverting. Now, let's review what we've learned up to this point. When we want to divide any number by a fraction, we first invert the fraction. That tells us how many times that fraction is contained in one. Then we multiply by the number of ones we have. That gives us our answer. Now we know our rule. When you wish to divide any number by a fraction, you invert your divisor and multiply. Now let's see how we might use our rule at home. Mary has two yards of ribbon. She wants to make some hair ribbons. She knows that each bow will require 2/3 of a yard. How many bows can she make from her two yards of ribbon? What Mary wants to know is how many times is 2/3 contained in two? To find the answer, she inverts her divisor and multiplies. The answer is three. Then she ties the bows. She finds that her arithmetic has given her the correct answer. Let's work another problem. Tommy is building a scale model of an airplane. He's now ready to assemble the wings. He finds that he needs four strips 3/16 of an inch wide to complete it. He has a piece of balsa wood 3/4 of an inch wide. He wonders how many strips 316 of an inch wide he can make from this. His problem is how many 316 are there in 3/4s? He writes his problem out like this. Then he inverts the divisor and multiplies. The answer is four. That would be enough. He now divides the large piece and it comes out exactly as his arithmetic told him it would. You can see that when you're making things, fractions can be a very useful tool. Fractions are easy to use when you understand the rules.