You have not gained anything, since you still have one fraction divided by another. Why do we usually not do this? Because I chose that problem carefully so the divisions would work out. We COULD do it just by dividing the numerators and denominators: The method of multiplying the reciprocal is usually the easiest. But is that really true? I played around with alternatives just to see what I could do, and replied: There are several ways to divide fractions, some more direct than others. What Alex meant, of course, was that you can’t just divide them as they stand, but have to change it to a division. I want to know WHY you cannot divide them. But before I get back to those, there is another question: Do I have to multiply by the reciprocal?įirst, a question (also from 1999) asking “Why can’t we?” triggers the answer, “Actually, we can!”: Dividing Fractions We have given many other explanations of this, both the reason for it and the process. Just remember: invert the fraction you are dividing by (the divisor), and multiply instead of dividing. If you are comfortable with multiplying fractions, then you will have no trouble dividing fractions. We can even replace the whole number in the problem with a fraction, like this: The same method always works, even if the answer isn't a whole number as in the simple examples. And that is the standard way to divide by a fraction. That amounts to multiplying by the reciprocal, 5/2. Here, in addition to multiplying by the denominator, 5, we had to divide by the numerator, 2. Then replace the division sign by a multiplication sign. The answer isĭo you see how the two equations are related? You take the fraction, 2/5, and "turn it upside down" (in math words, we say "invert the fraction," or "take its reciprocal"). Since the pieces we want are twice as big, there are only 1/2 as many of them. The bottom block is a different shape, but it's the same size, because it's made of 2 1/5-size pieces. Let's divide the 2 blocks into pieces this size. Here is 2/5 of a block (the shaded part): So, dividing by a fraction with 1 in the numerator amounts to multiplying by that numerator. In other words, we multiplied the number of pieces by 5. In other words,ĭo you see why? To make 1/5 size blocks, we had to make each whole block into 5 pieces. There are 10 pieces this size in the 2 blocks. To divide 2 by 1/5, we divide 2 blocks into pieces the size of the 1/5 block. Here is 1/5 of a block (the shaded part): What is 2 divided by 1/5? Here are 2 blocks: Once you understand them, you will be able to put them together to make harder problems. I've tried and cannot find a way.ĭoctor Rick answered Olivia’s question, starting with pictures of fractions in order to make the ideas concrete: Let's think about a few simple examples. How do you divide fractions? I love math and I am good at it. Here is a basic question, from 1999: Dividing Fractions Note: In this post, I am going to replace the slash (/), which we had to use on the old site, with the obelus (÷) when referring to division (as opposed to fractions), to make it easier to follow what we say. These show the reasons for the standard method, presented in a variety of ways, together with some alternative methods. Even looking only at division of fractions, I have had to restrict my attention to a few sample answers. Fractions have always given students trouble, and we have had many questions about working with them.
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