Fractions are my enemies! How do I make working with fractions easier?
1. Adding fractions
These two major shortcuts have been moved to addition tips
2. The "Gazintas" speed-up
type: speed-up
reliability: 
- has no limitations
understandability: 
- high
time saving: 
- big gain
usefulness: - absolutely a must
difficulty: - very easy
required skill: - beginner
overall: 43 of 48 points,
Many should know this handy trick, however, some do not. When you have multiplication, you should cross divide, as I call, "gazintas"*. Let's say you come across a situation in which you get:
4 7
- × - = ?
5 8
If you multiply it out normally, you'd get:
4 7 28 7
- × - = -- = --
5 8 40 10
... because 28/40 reduces. Do you see a clue? If you don't get it, there is a way to do it without simplifying it and working with much simpler numbers. Consider cross-GCF-ing [GCF stands for "greatest common factor"]. You can find that four goes into both successfully, so divide the 4 and the 8 to get 1 and 2 respectively. This changes the question to:
4 7 1 7 7
- × - = - × - = --
5 8 5 2 10
This is quicker only because you're working with simpler numbers. Who'd want to multiply the following without doing this right away?
128 54
--- × --- = ?
135 192
First off, you could reduce 64 from 128 and 192 to get 2 and 3 and reduce 27 from 135 and 54 to get 5 and 2, far simpler than having to multiply 128 and 54 then 135 then 192, even with applying those handy multiplication tricks I have available, using this trick would be so much faster. However, what if you decide to use the column multiplication trick where you multiply three numbers at once without setting anything aside? It works the same, only choose two numbers that can get reduced. Here's a good example:
5 2 3 1 2 3 1 2 1 1 1 1 1
- × -- × - = - × - × - = - × - × - = - × - × - = --
6 15 7 6 3 7 2 3 7 1 3 7 21
As you can see, there are plenty of steps. Normally, you'd have 30/630. Of course, you can drop the zeros on the end then simplify that.
This trick would be very handy and can be up to 60% faster. Who'd want to deal with the fraction 6912/25920 when it can be reduced to 4/15, much easier to understand, yet a common fraction. For dividing, it's exactly the same as multiplying, but there's two ways to speed it up. One is very slight, the other is the same as above. For division, don't cross-GCF, instead go across as you would in multiplication. For the same question mentioned with the large fraction directly above, you'd consider 128 and 54 rather than 128 and 192. In this case, you could only take two's out and the bottom two cannot be reduced. Then, multiply them like you would normally, only do it cross ways. If the number on the top is used with the number on the bottom, your answer goes on top, and the reverse for the other two.
3. Dividing fractions [general]
type: shortcut
reliability: - very high
understandability: - very high
time saving: - small gain
usefulness: 
- medium-high
difficulty: - easy-medium
required skill: - novice
overall: 36 of 48 points,
Dividing fractions requires an extra step, changing it to multiplication, but you don't need that neccessarily. If you're simply flipping the second fraction around, why don't you just cross-multiply instead? Yet, you can still presimplify the question without flipping it around. With multiplication, if you have something on the top that can be crossed over with something on the bottom, division is always straight across [unless multiplication is slipped in].
That's all the tricks I have for fractions at the moment.
Footnotes:
* There is a game that I created called "gazintas". To learn about it and how to play, refer to the games index.
