Dividing Fractions

Dividing fractions. Before we talk about dividing fractions and what that means, let's look at an ordinary whole number division problem. 6/2. The six, the number being divided is called the dividend. The two, the number being divided by is called the divisor. And the answer, the number of times the divisor can fit into the dividend is called the quotient. To nobody's surprise, two goes into 63 times. But let's use a number line to look at this a bit more closely. We'll represent the dividend in the problem with this bar and the divisor with this bar. Now, we'll see how many times two will fit into six. Sure enough, two still goes into 63 times. Let's do another problem. The dividend will be four a whole number. But this time, we'll throw a fraction into the mix and use one quarter as the divisor. Since we're now dividing by a fraction, we'll put up markers for one quarters on the number line, and we'll label the markers as how many fourths those quarter markers represents on the number line. Here's our dividend, four. And here's our divisor, one quarter. How many times will one quarter fit into four? If we count all those fourths or look at the quarter marks on the number line, we see that one quarter will go into 416 times. That's great. We got a way to visualize what's happening when we divide by fraction. In this case, we can see that for each whole number on the number line, we have four fourths. So we've got four groups of four, which is four times four, and that equals 16. Hey, would you look at that 4/1 quarter equals 16, and four times four also equals 16. There's something going on here, let's take a look. One fourth and four are what we call reciprocals. One quarter is the reciprocal of four and four is the reciprocal of one quarter. A whole number like four, can be written as a fraction by putting it over one. The whole number four is the same as four once. But reciprocals aren't just for when you have a number and one over that number. If you have a fraction like two thirds, to get the reciprocal, you just flip it to get three halves. Reciprocals are very special and useful numbers. There's a lot that could be said about them. But for our purposes here today, we're going to remember that they are very useful for division. But before I spill the beans on that, let's look at our last problem again. Looking at these two equations, we can see that dividing by one quarter gets the same result as multiplying by its reciprocal four. Is this something special about four, one fourth, and 16 or does it work with other numbers? And does it only work when dividing by fractions? Let's look at our first problem. 6/2 is three. There's not a fraction in sight, at least not yet. Is this true that six times one half the reciprocal of two is three. Six times one half is the same as adding one half six times. Every part of one halves is equal to one, which leaves us with three ones. But I'm still skeptical. Let's give it the number line treatment. Here's our number for 6/2 equals three. Let's try that with six times one half equals three. Six times one half is six halves. One, two, three, four, five, six. Okay, I'm sold. Six halves is three. So dividing six by two gives us the same results as multiplying six by the reciprocal of two. We could work a lot more problems and we get the same results. We've got a new rule for division. And here's the rule. Dividing by a number is the same as multiplying by its reciprocal. And it works both ways. Multiplying a number is the same as dividing by its reciprocal. Let's test it with a fraction division problem. A problem that's all fractions. Dividing a fraction by a fraction. I'm so excited, two thirds divided by one sixth. If we try and solve it the way we did with the number line, we can see that the 16 fits into two thirds four times. Okay, okay, I'm just checking. Here comes the reciprocals. Two thirds divided by one sixth is the same as two thirds times 6/1. And the answer is, let's see here. Two times six is 12. And three times one is three. That's 12 thirds. And 12 thirds simplifies to four. With this method, we can solve all sorts of fraction division problems. We can change the division to multiply and flip the divisor. Three times five is 15, four times two is eight. This gives us 15 eighths, which is an improper fraction. We can simplify that to one and seven eighths. Now let's divide a fraction by a whole number, two thirds divided by five. Change the operation from divide to multiply and flip the five to reciprocal one fifth. Two times one is two, three times five is 15, 2/15. And that's the story on dividing fractions. A common way to remember the steps is to say, keep, change, flip. Keep the dividend, change, divide to multiply, and flip the divisor.