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The problem 2 ÷ 1/2 is asking you: ”How many halves are in 2?”  The answer is 4, because each unit (1) is made up of two halves, and there are 2 units total: 2 halves/1 unit * 2 units = 4 halves.  Try thinking about this same equation in terms of cups of water: How many half cups of water are in 2 cups of water?  You could pour 2 half cups of water into each cup of water which means you are basically adding them, and you have two cups: 2 halves/1 cup * 2 cups = 4 halves. All of this means that when the fraction you are dividing by is between 0 and 1, the answer will always be larger than the original number!  This is true whether you are dividing whole numbers or fractions by a fraction. Therefore, dividing by a fraction can be accomplished by multiplying by its reciprocal.  The reciprocal of a fraction (also called its “multiplicative inverse”) is just the fraction turned upside down, so that the numerator and denominator have switched places. In a moment, we are going to divide fractions by fractions by finding the reciprocal of the second fraction and multiplying them together, but let's look at some reciprocals first:  The reciprocal of 3/4 is 4/3. The reciprocal of 7/5 is 5/7. The reciprocal of 1/2 is 2/1, or 2. In order, the steps are:  Leave the first fraction in the equation alone. Turn the division sign into a multiplication sign. Flip the second fraction over (find its reciprocal). Multiply the numerators (top numbers) of the two fractions together.  This result will be the numerator (top portion) of your answer.    Multiply the denominators (bottom numbers) of the two fractions together. The result will be the denominator of your answer. Simplify your fraction by reducing it to the simplest terms. We will begin by leaving the first fraction alone, and changing the division sign to a multiplication sign:   1/3 ÷ 2/5 = becomes: 1/3 * __ = Now we flip the second fraction (2/5) over to find its reciprocal, 5/2: 1/3 * 5/2 = Now multiply the numerators (top numbers) of the two fractions, 1*5 = 5. 1/3 * 5/2 = 5/ Now multiply the denominators (bottom numbers) of the two fractions, 3*2 = 6. We now have: 1/3 * 5/2 = 5/6  This particular fraction cannot be simplified further, so we have our answer. "Dividing fractions, as easy as pie, Flip the second fraction, then multiply. And don't forget to simplify, Before it's time to say goodbye."  Another  helpful saying that tells you what to do with each part of the equation is: “Leave Me (the first fraction), Change Me (the division symbol), Turn Me Over (the second fraction).”
Think about what dividing by a fraction means. Understand that dividing is the opposite of multiplying. Memorize the following steps for dividing a fraction by a fraction. Work through these steps on the example 1/3 ÷ 2/5. Try remembering the following rhyme to help you remember: