Multiplying 3 numbers at once to find age in seconds at 21 years

Fractions are my enemies! How do I make working with fractions easier?

Last updated: around August 2005

1 Adding fractions

These two major shortcuts have been moved to addition tips

2 The "Gazintas" speed-up

Type:              Speed-up
Reliability:       5 stars - Extremely high
Ease of learning:  4.5 stars - Very easy
Time saving:       4.5 stars - Large gain
Usefulness:        5 stars - Extremely useful
Difficulty:        4.5 stars - Very easy
Overall:           4.5 stars - Excellent

2.1 The shortcut

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.

2.2 An extreme example of using the shortcut

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. What I usually do, however, is keep taking out 2's from both parts until I no longer can. I then go to 3's, 5's, 7's, 11's, and continue on with prime factors until I'm left with primes or no further way to reduce a number (like a 15 and 7, though 15 isn't prime (it's got a 5 and 3)). Don't forget to reduce the fractions if they can (such as reducing the 54/192 into 27/96, taking 2 out).

2.3 Muliplying multiple fractions

What if you decide to use column multiplication where you multiply three or more numbers at once without setting anything aside? It works the same, only choose two numbers that can get reduced and keep reducing until nothing can be reduced any more. When fully reduced, just multiply. 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.

2.4 Dividing fractions

Type:              Shortcut
Reliability:       4.5 stars - Very high
Ease of learning:  4.5 stars - Very easy
Time saving:       4.5 stars - Large gain
Usefulness:        4.5 stars - Very useful
Difficulty:        4.5 stars - Very easy
Overall:           4.5 stars - Excellent

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.

* There is a game that I created called "gazintas". To learn about it and how to play, refer to the games index.