Dividing Fractions With Pictures!

(Also see Multiplying Fractions With Pictures!)

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During a recent interview, I was asked to show (1/2) ÷ (1/3) with pictures, and, although I had done similar work in grad school, I was a bit unsure of my answer.  Explaining why we multiply fractions the way we do is a much easier task.  Being put on the spot can sometimes be a good thing!

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IMPORTANT: We always want to remember that dividing is asking “how many of that fit into this?“.  In 10 ÷ 2, we are really asking “how many 5′s fit into 10?”  This is the same question we ask when dividing fractions: “how many of the second fraction fit into the first?”

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Let’s say that we wanted to find (2/5) ÷ (1/8).  Convention tells us to “flip the 1/8 and multiply across”.  But can we show this with pictures?

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This picture below represents 2/5.

Out of the 5 rows we have 2 orange ones.

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Now to address “divide by 1/8″ we want to divide the entire area into 8 vertical columns (8ths)…. 

… and take 1 of these columns (because we want 1/8)….

The darker line surrounds 1 out of the 8 vertical columns we had drawn in the previous step.  This area – 1×5 or 5 – will be our answer’s denominator.  But what will the answer’s numerator be?

In the above picture, the “extra” orange boxes have been colored green to show that they are outside the bounded area.  We now have to figure out how many sets of 5 (from area 1×5) we have. 

We have 3 sets of 5 and 1 left over, or 3 and 1/5.  This answer’s improper fraction is 16/5, which is the number of colored boxes over the total number of boxes within the bounded 1×5 region.

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We can also solve fraction division problems where the second fraction’s numerator > 1, as in (2/5) ÷ (3/8).  Here, we will look at the problem in a slightly different way than we did above.  Do you notice the difference?  Why do we still get the correct answer?-

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Here we see 2/5 again.

 Out of 5 rows, we have 2 green ones, representing 2/5.

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To address the “divide by 3/8″, we then need to divide the entire area into 8 vertical columns (8ths)…. 

… and take 3 of these columns (because we want 3/8)….

The darker line surrounds 3 out of the 8 vertical columns we had drawn in the previous step.  This area – 3×5 or 15 – will be our answer’s denominator.  Now to find the numerator…

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We now have to see how many colored boxes (numerators) will fit into our 3×5 bounded area…

Moving the 10 blue boxes into the blank ones within the 3×5 bounded area, we get….

…15 of the 15 boxes are colored in, and there is 1 extra colored box.  15 and 1 left over, or 1 and 1/15  (ie: 16/15: see SHORTCUT).

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SHORTCUT: Yes, it also works to count the colored boxes (numerator) and count the number of boxes within the bounded area (denominator) to get 16/15.  Cheater!  :-)

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Hope this helps!

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