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May 2013

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Division of Fractions

This topic is often not understood real well by teachers or students. But we want them to learn, not only the rule, but also the meaning.

These ideas can help you to explain and understand division of fractions:

1. The rule of "invert and multiply" applies to division in general - not just to division of fractions. It is a general principle. For example:
   
   20 ÷ 4
   
   I can invert and multiply:
   
   20 × 1/4 = 5.
   
   With whole numbers, division can be thought of as making equal parts. When you divide something by 7, you're dividing it into 7 parts, so might as well just take 1/7 part - multiply by 1/7.
   
   You can always change division into multiplication with this principle: 18 ÷ 2.51 = 18 × 1/2.51
    
2. Think of fraction division this way: how many times does the divisor fit into the dividend? You can use this to judge the reasonableness of your answer.
   
   For example consider 1 3/5 ÷ 2/3. Clearly 2/3 can fit into 1 3/5 more than two times.
   
   1 3/5 ÷ 2/3 = 8/5 × 2/3 = 16/15 = 1 1/15 - just a tad over 1 whole. CAN YOU SPOT THE ERROR?????
   
   Or, 3/8 ÷ 11/12. Here the divisor is greater than dividend. Well, that means that it won't fit even once into 3/8; it only "fits" into 3/8 about half ways, so the answer should be near half.
   
   And indeed, using the rule, 3/8 ÷ 11/12 = 3/8 × 12/11 = 3/2 × 3/11 = 9/22.
    
3. One alternative method for fraction division is to first change both the dividend and divisor into equivalent fractions, and then simply divide the numerators.
   
   5/6 ÷ 1/8 =
   20/24 ÷ 3/24
   = 20 ÷ 3 = 6 2/3.
   
   The answer makes sense since 1/8 can "fit" more than six times into 5/6.
   
   I think this is neat, because it helps make sense of the procedure: how many times can 3/24 fit into 20/24? It's the same as asking how many times can 3 fit into 20.
   
   Another:
   2 2/11 ÷ 2/5 = 24/11 ÷ 2/5
   = 120/55 ÷ 22/55
   = 120 ÷ 22 = 5 10/22 = 5 5/11.
   
    
4. Back to the rule of "invert and multiply". Let's think about number 1 as the dividend first.
   
   How many times does 1/2 fit into one? Two times. 1 ÷ 1/2 = 2.
   
   How many times does 3/4 fit into one? It fits there once, and there's 1/4 left, and into 1/4 we can fit 1/3 of 3/4. So total 4/3 times.
   1 ÷ 3/4 = 4/3.
   
   How many times does 1 2/5 fit into one? Not even once, clearly. But if you think of 1 has 5/5, you can see that five of the 7 fifths can fit... so 5/7 times. You might have make a picture of this in your mind or paper. Draw one whole as 5/5, then draw 7/5 next to it. Exactly five of the 7 parts of 7/5 fit into one.
   
   1 ÷ 7/5 = 5/7.
   
   You can follow this thinking with any fraction m/n: 1 ÷ m/n = n/m.
   
   OK, so if 5/6 goes to one exactly 6/5 times, then how many times can 5/6 fit into 3 13/15 ?
   
   Exactly 6/5 × 3 13/15 times.
   
   Or, 3 13/5 × 6/5, if you like. Invert and multiply!