Visual Division of Non-Unit Fractions
My article on Visualising the Division of Fractions is spreading like wild fire on social media. The basic idea is that when you divide by a fraction (e.g. half), we are counting with smaller units (e.g. the halves), so we get proportionally more in number (e.g. double) to compensate. This is why we multiply.
Did you notice that I purposely chose simple examples to ease everybody into that idea? All the examples I gave were divisions by unit fractions, which are fractions with 1 as numerator.
What if the numerator is not 1? This is what we are going to discuss in this article.
Example 1
Let us say you want to know the meaning of . To visualise this, we draw 4 bars. Since we are dividing by two-thirds, let us subdivide each of the 4 bars into 3 equal smaller pieces (thirds).
What does dividing by two-thirds mean? You take every two of smaller pieces (thirds) and group them and treat that as your new unit (indicated in red in the diagram). So how many groups of two smaller pieces are there? Notice that every 2 original whole units comprises 6 thirds, and that forms 3 groups of two small pieces. Observe that
(1) The ratio of the size of each new unit compared to the size of each old unit is 2:3.
(2) The ratio of the number of the new units to the number of old units is 3:2, the inverse ratio.
And that is why the number 4 of original (blue) units is multiplied by .
You have smaller units (red) than the original units (blue). To compensate, the number of smaller units is proportionately more.
Example 2
Let us look at a slightly harder example. Let us say you want . You pre-divide each original unit (shown as blue pentagons) into 5 equal smaller triangular pieces. Each small piece is one-fifth of the pentagon.
Since we are dividing by three-fifths, we take every group of 3 of the small pieces as our new unit, and then we count the number of such groups. It can be a little tricky, as we might need to take small pieces from another pentagon to form a group of three pieces. But it can be done. Every 3 pentagons (blue) give rise to 5 new groups (red), as they both contain 15 small triangular pieces. So we see that
(1) The ratio of the size of each new unit compared to the size of each old unit is 3:5.
(2) The ratio of the number of the new units to the number of old units is 5:3, the inverse ratio.
And that is why the number 6 of original (blue) pentagons is multiplied by .
You have smaller units (red groups) than the original units (blue pentagons). To compensate, the number of smaller units is proportionately more.
Example 3
For our pièce de résistance (or is it coup de grâce?), we shall consider division by a fraction greater than 1. For instance, let us look at . Actually this is easy. If we have 6 circles, say, we first sub-divide them into halves.
Division by 3/2 just means every group of 3 halves forms a new unit (shown in red). Then we just determine the number of such groups. Every 3 circles, constituting 6 halves, gives rise to 2 such groups. Hence
(1) The ratio of the size of each new unit compared to the size of each old unit is 3:2.
(2) The ratio of the number of the new units to the number of old units is 2:3, the inverse ratio.
And that is why the number 6 of original (blue) circles is multiplied by .
You have larger units (red groups) than the original units (blue circles). To compensate, the number of red groups is proportionately less.
In general, we have
And this is the reason why you flip the fraction. The principle still remains: If you have larger new units, there will be proportionally fewer new units. If you have smaller new units, there will be proportionally a greater number new units. Since the relevant proportions are inverses of each other, the number of new units can be calculated via multiplying the reciprocal (inverted fraction) of the divisor.
So teachers, parents, students, citizens of the world!
Now you know the reason why,
Therefore invert and multiply!
Cheers!
Announcements
If you find my articles useful or interesting, please upvote and resteem them! Thanks !
Please join my Sharp Mathematics Contest 2, which is based on the discussion in this article. Resteem and tell all your friends about it!
Cheers!!!
@tradersharpe
-- promoting sharp minds
This gem of a post was discovered by the OCD team!
Reply to this comment if you accept, and are willing to let us promote your post!
If you accept this, you'll be nominated and the members of the OCD team will vote on whether we'll feature your post in our next compilaton.
You can follow @ocd - learn more about the project and see other gems! We strive for transparency.
I'm mostly curating in #science and am always glad to see quality posts like yours peeking out between the garbage and plagiarism!
thank you! please do resteem and publicise! you are not only helping me, you are raising human consciousness! ;-)
@minnowpond1 has voted on behalf of @minnowpond. If you would like to recieve upvotes from minnowponds team on all your posts, simply FOLLOW @minnowpond.
@cmtzco has voted on behalf of @minnowpond. If you would like to recieve upvotes from minnowponds team on all your posts, simply FOLLOW @minnowpond.
To receive an upvote send 0.25 SBD to @minnowpond with your posts url as the memo
To receive an reSteem send 0.75 SBD to @minnowpond with your posts url as the memo
To receive an upvote and a reSteem send 1.00SBD to @minnowpond with your posts url as the memo
img credz: pixabay.com
Steem is behaving weird in low voting percentages, so actual votes can diverge by up to 1 voting percent point.
Nice, you got a 3.0% @minnowbooster upgoat, thanks to @tradersharpe
Want a boost? Minnowbooster's got your back!
The @OriginalWorks bot has determined this post by @tradersharpe to be original material and upvoted it!
To call @OriginalWorks, simply reply to any post with @originalworks or !originalworks in your message!
To enter this post into the daily RESTEEM contest, upvote this comment! The user with the most upvotes on their @OriginalWorks comment will win!
For more information, Click Here! || Click here to participate in the @OriginalWorks sponsored writing contest(125 SBD in prizes)!!!
Special thanks to @reggaemuffin for being a supporter! Vote him as a witness to help make Steemit a better place!