Fastest way you have found to multiply two numbers in your head?

In summary, the fastest way to multiply two numbers in your head is to use the subtraction method or the grouping method. These methods involve breaking down the numbers into smaller, easier to work with parts and then adding them together. By choosing round numbers and using shortcuts, it is possible to multiply larger numbers quickly and efficiently in your head. However, these methods may require some practice and familiarity with numbers in order to be effective.
  • #1
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This is a mental arithmatic question really, but what is the fastest way you have found to multiply two numbers in your head?

How many steps does it take you and what shortcuts do you use?

I apologise if this is the wrong forum.

Examples:
33x24
123x321
9999x6666
etc..
 
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  • #2
Wow, do you all consider this too simple a question to answer?
 
  • #3
Well the first one is perhaps most easily calculated as 660+132=792
the last one as 66660-6666=60000-6=59994
 
  • #4
I thought the first was better calculated as 720 + 72 = 792.
 
  • #5


Maybe a better description of what they are trying to say is by using the distributive property.

Ex. 33(24) = 33(20 + 4) = 660 + 132 = 792, you could also have done
24(30 + 3) = 720 + 72 = 792. When you do it often you pick up which is easiest for you sometimes people are better at certain multiples like 6's versus 8's. Also, don't forget it works over subtraction.

Ex. 66666(99999) = 66666(100000 - 1) = 6666600000 - 66666 = 6666533334, although I highly doubt that many people would be able to do it quickly in their head, but its still faster that way then doing via the method you learn in school.

Also, you can split both numbers, but that's rarely as efficient because of the increased amount of work you'd need to do since you would FOIL.

Ex. 33(24) = (30 + 3)(20 + 4) = 600 + 120 + 60 + 12 = 792
 
  • #6


This isn't the most practical way to multiply, and it certainly isn't something you'll probably use, but I thought it was cool.

http://www.youtube.com/watch?v=otCLgQjBaio&feature=related

By the way, has anyone ever seen this before? I'm interested in why it actually works. It's youtube so the video comments were no help.
 
  • #7


What is convenient when doing computations in your head is to get rid of the number of carry digits that you need to keep track of even if that translates to a method that looks less efficient from an algorithmic point of view. You can also subtract a round number from both numbers to simplify the multiplication.

The subtraction method works as follows. We want to compute a*b. If N is some arbitrary number, we can define:

a' = a - N

b' = b - N

Then we have:

a*b = (N + a')(N + b') =

N^2 + N(a' + b') + a' b' =

N(N + a' + b') + a' b' =

N( a + b') + a' b'


The trick is to choose N such that it is an easy round number to work with, while a' and b' are small and/or round. If a and b are large numbers, then the best you will be able to do is get a' and b' that are an order of magnitude less than a and b. But then you can iterate this procedure using some other round number M to write:

a' b' = M(a' +b'') + a'' b''

where a'' = a' - M and b'' = b' - M

Multiplying in this way requires more steps than your calculator uses, however the difficulty you face when doing computations in your head is not really a lack of computing power, as your brain has vastly more computing power than the most powerful supercomputer, it is simply that you only have access to some limited functions of your brain to do arithmetic. So, what you must do is make sure you can easily keep track of numbers that appear in the various stages of the computation.


Simple example:

998 x 983

Subtract 1000:

998 x 983 = 1000 x 981 + 2x17 = 981000 + 34 = 981034

Ok, this was a contrived example, let's look at a more realistic example:


538 x 721

Let's subtract 500:

538 x 721 = 500x(721 + 38) + 38x221 =500x759 + 38x221

500x759 = 500x760 - 500 = 760000/2 - 500 = 380000 - 500

To compute 38x221, let's subtract 40:

38 x 221 = 40x(221 - 2) - 2x181 = 40x(220-1) - 362 =

8800 - 402

So, the answer is:

380000 - 500 + 8800 - 402 = 388000 - 100 - 2 = 387898

So, even this could be done in your head as the computation only involves easy to work with round numbers.

If you multiply directly without any tricks, you can group together the same powers of ten to minimize the number of carry digits.

So, if we multiply

[c3*10^3 + c2 * 10^2 + c1*10 + c0]x
[d3*10^3 + d2 * 10^2 + d1*10 + d0]

you should compute the digits of this multiplication by computing the last digit first from c0*d0, remember the carry digit.

Then you evaluate c1*d0 + d1*c0 and add the previous carry digit to find the next digit and remember the new carry digit. then you evaluate c2*d0 + d1*c1 + d2*c0 and add the previous carry digit, etc. etc. This way, you only have to remember one carry digit in each step, and the digits of the answer. With some practice you can multiply two five digit numbers in your head this way.
 
  • #8


The subtraction method can be generalized by subtracting some small multiple of N:

If we put

a' = a - p N

b' = b - q N

we get:

a*b = (p N + a')(q N + b') =

N(q a + p b') + a' b'


Example:

853 x 238

Subtract multiples of 200:

853 x 238 = 200[853 + 38x4] + 38x53

853 + 38x4 = 853 + 160 - 8 = 1005

853 x 238 = 200x1005 + 38x53 = 201000+38x53

Compute 38x53 by subtracting 40:

38x53 = 40x51 - 26 = 2040 - 26 = 2014

So, we have:

853 x 238 = 201000 + 2014 = 203014
 

1. What mental strategies do you use to quickly multiply two numbers in your head?

To quickly multiply two numbers in my head, I use the following strategies:

  • Break down the numbers into smaller, easier to multiply parts.
  • Use known math facts, such as multiplication tables, to simplify the calculation.
  • Round the numbers to the nearest multiple of 10 to make the calculation simpler.
  • Use the distributive property to break down more complex multiplication problems.

2. How can I improve my mental math skills for multiplying two numbers?

To improve your mental math skills for multiplying two numbers, you can:

  • Practice regularly with different numbers and types of multiplication problems.
  • Memorize multiplication tables and common multiplication patterns.
  • Use mental math games and exercises to strengthen your skills.
  • Try using different strategies and find the ones that work best for you.

3. Is there a specific order in which I should multiply the digits of two numbers?

There is no specific order in which you should multiply the digits of two numbers. However, it can be helpful to start with the larger digits and work your way down to the smaller ones. You can also try breaking down the numbers into smaller parts and multiplying them separately before combining the results.

4. How do you handle carrying and borrowing when multiplying two numbers in your head?

When multiplying two numbers in my head, I try to avoid carrying and borrowing by breaking down the numbers into smaller parts. However, if carrying or borrowing is necessary, I use mental math strategies such as rounding or reorganizing the numbers to make the process easier.

5. Can you provide an example of using mental math to quickly multiply two numbers?

Sure, let's take the problem 36 x 27 as an example. First, I would break down the numbers into smaller parts: 36 can be broken down into 30 + 6 and 27 can be broken down into 20 + 7. Then, I would use the distributive property to multiply each part separately: (30 x 20) + (30 x 7) + (6 x 20) + (6 x 7). Using known math facts, I can quickly calculate the individual parts: 600 + 210 + 120 + 42. Finally, I would add these numbers together to get the final answer of 972. This may seem like a lot of steps, but with practice, it becomes second nature and can be done quickly in your head.

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