What kinds of addition speed-ups and shortcuts do you have?
1. The nine and scrap trick
type: speed-up
reliability: 
- medium-low
understandability: 
- medium-high
time saving: 
- small gain
usefulness: - moderately high
difficulty: 
- very easy
required skill: - beginner
overall: 33 of 48 points,
The nine and scrap trick can speed up addition by about 10 to 20%. The trick has limitations, however, it's title lists two. Let's consider this simple addition question:
993
+ 8
Just about anyone could add that baby question, however, you can actually speed it up. Worked out normally, you'd have:
11
993
+ 8
1001
However, to apply the trick, three conditions must be met:
- You have a 9 somewhere in the number except the digit to the far right.
- You had to carry in the previous step before a 9 is used. It may be a nine on the previous step, but the next digit if you carry, must be a nine regardless on what was before it.
- This trick only applies to two numbers being added, not a column of numbers.
- There are no restrictions whether whole numbers or decimals are used.
Look at the original question again. Note that you had to carry on the first step. Because a 9 was involved in the next column, you can apply the nine and scrap trick. This rather complicated addition question would go by in a breeze using this trick:
24959997198902907922989999936798999149
+ 9294719949299499399499497199919090992
On every step with the first, there is a 9 involved in every step. Worked out normally, it may take about 40 seconds on the high end to work it out, but using the nine and scrap trick, the only skill you need is copying and identifying and can be done in half the time! Here's how to apply the trick:
- Work the question out until you have to carry.
- Check to see if a 9 is in either row. Two 9's means that there is a 9, thus appliable.
-
- If there isn't a 9 in either row, add normally.
- If there is a 9, just copy the other digit down and carry the 1.
-
- If the current step doesn't have a 9, add normally.
- If the current step has a 9, repeat the 9 and scrap trick until the current step doesn't have a 9.
This rather complicated question involves the nine and scrap trick all the way through. If decimals are involved, the trick is just as effective only with the usual added step: Line up the decimals then add applying the nine and scrap trick whereever possible. If you have multiplication, and that addition is needed and that only a pair of numbers needs to be added, the nine and scrap trick is effective here as well. Consider these two questions, one multiply [almost completely worked out so you can see a case of where the nine and scrap trick would be of use], ther other one deals with adding decimals:
215
327
1428 26989.979
× 79 + 2694.09
12852
+99960
This, once you get used of it, can speed up pencil-and-paper addition by 10 to 20%, depending on how fast you realize the situation.
2. The plus-minus fraction multiply trick
type: shortcut
reliability: - high
understandability: 
- medium
time saving: - big gain
usefulness: - high
difficulty: - easy-medium
required skill: - novice
overall: 34 of 48 points,
This trick involves fractions. Adding and subtracting fractions, to most, especially with unlike denominators*, can be quite a challenge. However, there are two very handy shortcuts dealing with fractions.
The first involves odd denominators, though it works with any denominator, even if they are alike. Picture this simple situation:
If you had a 3/4 cup and a 2/3 cup, how much bigger is the 3/4 than the 2/3?
3 2
- - -
4 3
Here, you are subtracting, to find the difference. The usual way involves having to find the least common multiple, however, there is a shortcut to avoid this extra step, or at least, most of it. The old way is to count by the number listed so far then do the other. Then find the lowest match:
4 8 12 16
3 6 9 12
Right away, you see that 12 appears first in both lists. That is the denominator you use. Then you try to find equal fractions, again, another useful shortcut is available. This changes the question to:
3 2 9 8
- - - = -- - --
4 3 12 12
From this point, you can easily answer the question. However, you can almost cut the time required in half! There are two available shortcuts. The first one eliminates the counting-by-the-numbers-to-find-a-match method. The second one eliminates the need to having to try to find equal fractions. First, let's go through this same question again, only with using the shortcuts, both can be used in the same question, regardless of it's type!
Take a look at the two denominators. Then take a look at what you get. Do you see a clue? If you can't, try putting some operation between the 4 and the 3 and see. If you still can't see it yet, note that 4×3=12. See the pattern? Just simply multiplying the two denominators will instantly give you a common multiple, but not always the least common multiple. To guarentee a least common multiple, there is a trick you can use to ensure it, though it might be confusing at first. This is explained after the two shortcuts are explained. If the numbers in the denominators are both prime, a least common multiple is always guarenteed.
The second tip is also quite straight forward but a bit more confusing to explain. If you wanted a least common multiple, follow the extra steps following a bit later. Here, you cross-multiply to find the equal fractions. The numerator is always the key to the location. That is, if the numerator used is in the first fraction, the numerator of the converted fraction is the first fraction. That is, Since 3×3 is 9, and since the first fraction contains the key, the result of 3×3, or 9, goes on the first fraction. For the case of 4×2, since the 2 is in the numerator of the second fraction, the result goes on the second converted fraction.
Then, subtract the easier versions with the shortcuts taken. However, what if you had this instead:
5 5
- + -
8 6
This approach requires an extra step, though the two shortcuts can still be used. Normally, using the old method, you'd get:
8 16 24 32
6 12 18 24
So you're least common multiple is 24 as it appears in both lists. Using the new method, 8×6 is 48, but 24 is the least common multiple. However, notice another pattern. 8 and 6 have 2 in common; they are both multiples of 2. If you temporarily divide either one [one only, not both] by 2, you can get either 4×6 or 8×3, both equal to 24. See the shortcut? On paper, I recommend that you place the new value underneath faintly so you won't confuse it with something else.
The second stage also requires this first step before multiplying. When it comes to cross multiplying, the denominator, must be divided by their common multiple, two, in each step. That is, when you multiply 5×6, you change the 6 to a 3 temporarily to get 5×3, or 15. 15/24=5/8. Repeat for 8×5. 4×5 is 20 and 20/24=5/6. Now you got your fractions. Add or subtract normally.
I haven't really thought much into adding multiple fractions, though it also works with subtraction as well.
3. The fraction to decimal trick
type: shortcut
reliability: 
- very low
understandability: - medium-high
time saving: 
- very big gain
usefulness: - medium
difficulty: - easy-medium
required skill: - beyond advanced
overall: 26 of 48 points,
This trick is a bit harder to pull off, but masters at arithmatic would find this as a shortcut to doing math in their head. This trick also has many limitations, depending mostly on your skill. Let's say you were adding:
3 1
- + -
4 5
Yeah, the tricks above can be applied, however, you can go faster yet. If you're good with fraction-decimal conversions, this trick can nearly double the speed of the standard way! First, you convert each fraction into decimals. 3/4=.75 and 1/5 =.2. Then, simply add the decimals. 3 steps versus 5 steps using the old, much slower way, and 4 steps using the simple method [as this would apply]. This method is also easier to work out in your mind than it is to with the other method. However, when repeating decimals are involved, especially from fractions like 5/7, 2/3, or 5/6, this trick can be very difficult to work out. Just multiply the denominators to find a common multiple then add the best you can in your mind. This is the method I use, though, working with 7ths is rather tough, let alone 17ths with 16 repeating digits....
Footnotes:
* Telling the difference between numerators and denominators can be tough. The best way is to think of denominators as down, both starting with the same letter and denominators are the bottom part of a fraction. The numerator is just the other one that remains.
