Multiplying 3 numbers at once to find age in seconds at 21 years
What addition shortcuts do you know of?
Last updated: around August 2005
Addition isn't one of my favorites, but I do happen to have shortcuts available, especially shortcuts with adding and subtracting fractions.
1 The nine and scrap shortcut
1.1 The concept
Reliability: - Medium-low
Ease of learning: - Easy
Time saving: - Medium gain
Usefulness: - Somewhat useful
Difficulty: - Very easy
Overall: - Good
The nine and scrap shortcut can speed up addition by about 10 to 20%. The shortcut has limitations, however, it's title lists two. Let's consider this simple addition question:
Just about anyone could add that baby question, however, you can actually speed it up. Worked out normally, you'd have:
However, to apply the shortcut, three conditions must be met:
- You have a 9 somewhere in one or both of the numbers being added 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 shortcut 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 shortcut.
1.3 A good example of a use
This rather complicated addition question would go by in a breeze using this shortcut:
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 shortcut, the only skill you need is copying and identifying and can be done in half the time! Be careful, however, when doing this, especially at high speed.
1.4 How to apply and use
Here's how to apply the shortcut:
- 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 shortcut until the current step doesn't have a 9.
This rather complicated question involves the nine and scrap shortcut all the way through. If decimals are involved, the shortcut is just as effective only with the usual added step: Line up the decimals then add applying the nine and scrap shortcut 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 shortcut 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 shortcut would be of use], the other one deals with adding decimals:
× 79 + 2694.09
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 shortcut
Adding fractions, especially with big double or triple-digit denominators can be a pain using the least common multiple technique. This shortcut virtually eliminates the need of doing this making triple-digit (or even 10-digit) denominators on paper go by nearly 10 times faster and faster yet. This super shortcut can turn 15 minutes of paper calculations into just 1 minute (based on someone who is fairly fast at doing math)! It's an instant 5-star rating for time! The ratings are based on the primary concept, not the case of guaranteeing the least common denomiator.
Reliability: - Extremely high
Ease of learning: - Very easy
Time saving: - Huge gain
Usefulness: - Extremely useful
Difficulty: - Very easy
Overall: - Superior
2.1 The old way
This shortcut 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?
- - -
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. This is highly disadvantageous when it comes to big denominators that could end up having a list being 120 numbers long for just a pair of two-digit denominators. 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, even for single-digit denominators!
2.2 Denominator multiplication
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! This shortcut becomes far more beneficial and useful when bigger denominators are involved.
Take a look at the two denominators in the original question, 4 and 3. Then take a look at what you get after doing it the old way, a 12. 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 shortcut 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.
2.4 Guarenteed least common multiples
Guarenteeing a least common multiple can be tricky with big denominators. However, what if you had this instead:
- + -
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 (but readable of course) 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 shortcut
This shortcut is especially decent for mental math, but it has big limitations, especially in terms of skill and when it comes to repeating decimals (like 7ths, 17ths, or 11ths).
Reliability: - Low
Ease of learning: - Easy-medium
Time saving: - Large gain
Usefulness: - Moderately useful
Difficulty: - Easy-medium
Overall: - Good
This shortcut is a bit harder to pull off, but masters at arithmatic would find this as a shortcut to doing math in their head. This shortcut also has many limitations, depending mostly on your skill. Let's say you were adding:
- + -
Yeah, the shortcuts above can be applied, however, you can go faster yet. If you're good with fraction-decimal conversions, this shortcut can nearly double the speed of the method in section 2! First, you convert each fraction into decimals. 3/4=.75 and 1/5 =.2. Then, simply add the decimals and convert back to a fraction again. 0.95 is 19/20. 4 steps versus 5 steps using the old, much slower way, and 4 steps using the simple method [as this would apply]. This shortcut is best used for mental math calculations rather than with paper and pencil. However, when repeating decimals are involved, especially from fractions like 5/7, 2/3, or 5/6, this shortcut 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....
* 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.
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