Base Method for multiplication


In this tutorial, we are going to learn the shortcut on how to multiply any 2 numbers which are closer to the powers of 10 like 10, 100, 1000 etc. This technique is called Base method for Multiplication.

To begin with, let’s say we want to multiply 12×13. Notice that both the numbers 12 and 13 are closer to the base 10. So we can use this shortcut. Let’s understand the steps involved.

 

BaseMultiplication_1

 

Step 1 :  Write the numbers 12 and 13 one below the other.  12 is +2 more than 10(base) and 13 is +3 more than the base. Write these offsets next to the numbers.

Step 2 :  Add the top left number ( 12 in this case) to the bottom right ( +3 the offset) or vice-versa. Either step results in the same answer. Adding 12 and 3 gives 15, which forms the L.H.S part of the answer.

Step 3 : Multiply the offsets(+3 and +2) i.e., 6 which forms the R.H.S part of the answer .

So, the final answer is 156. So, that’s it. No complicated multiplication.

Easy 🙂 ?

Important In the above example, we have used the base as 10 which has 1 zero. So when we multiply the offsets, we need to keep only 1-digit and the rest of the digits are carry forward.

 

Let’s consider another similar example. Can you compute what is 106 x 108?

Here, both the numbers are closer to the base 100. Notice how easy it is to get the answer, without actually multiplying the numbers.

 

BaseMultiplication2

Important : In the above example, we have used the base as 100 which has 2 zero. So when we multiply the offsets, we need to keep only 2-digit and the rest of the digits are carry forward.

 Now, consider this example, 104 x 102.

 

BaseMultiplication3

 

Note the offsets 4 and 2 when multiplied results in 8. However, as the base is 100 with 2 zeros, we need to have 2-digits in the R.H.S. So, we write the R.H.S part of the answer as 08. So the final answer is 10608.

If the base is 10, the R.H.S part of the answer should have only 1 digit. The rest of the digit will be carry-forward. If the base is 1000, the R.H.S part of the answer should have 3 digits and so on.

 

Let’s consider an example with a carry forward.  Consider the multiplication 115 x 107.

Both the numbers are close to 100. Now the offsets, 15 and 7 when multiplied results in 105. However, we can keep only 05 as the R.H.S part of the answer and 1 is carry-forward. The final answer would then be 12305.

 

BaseMultiplication4

 

On the similar lines, can you mentally multiply?

  • 15×12
  • 112×110
  • 113×108
  • 1004×1016

Multiplication of numbers when the numbers are below  base

 

So far, we only considered the numbers which are above the base. The same shortcut can be applied if the numbers are below the base. Consider this example : 85 x 87.

Notice, both the numbers are below the base( 100 in this case).  Let’s understand the steps involved.

 

BaseMultiplication5

 

Step 1 :  Write the numbers 85 and 87 one below the other.  85 is 15 less than 100(base) and 87 is 13 less than the base. Write these offsets next to the numbers. Notice the “-” sign in the offsets. This is to indicate that the numbers are less than the base.

Step 2 :  Add the top left number 85 to the bottom right ( -13 the offset) or vice-versa. Either step results in the same answer. Adding 85 and -13 gives 72, which forms the L.H.S part of the answer.

Step 3 : Multiply the offsets(-15 and -13) i.e., 180. However, we can keep only 80 as the R.H.S part of the answer and 1 is carry-forward. The final answer would then be 7395.

 

Can you now multiply 95 x 92?

 

BaseMultiplication6

Easy 🙂 ?

Similarly, try to  multiply mentally ?

  • 89×88
  • 96×95
  • 995×992

 

Multiplication of numbers when the numbers are above and below the base

 

So far, we saw the shortcut to multiply the numbers when both the multiplier and the multiplicand are above and below base. We can also numbers, where multiplicand is above base and multiplier is below base or vice-versa. Consider this example : 85 x 104.

Notice, here the multiplicand is below base(100) and the multiplier is above base.  Let’s understand the steps involved.

BaseMultiplication7

 

Step 1 :  Write the numbers 85 and 104 one below the other.  85 is 15 less than 100 and 104 is 4 more than base. Write these offsets next to the numbers. Notice the “-” sign in the offset. This is to indicate that the numbers are less than base.

Step 2 :  Add the top left number 85 to the bottom right ( +4 the offset) or vice-versa. Either step results in the same answer. Adding 85 and +4 gives 89, which forms the L.H.S part of the answer.

Step 3 : Multiply the offsets(-15 and +4) i.e., -60. However, we cannot keep -80 as the R.H.S part of the answer.

Step 4 : So we need to expand the answer obtained in the L.H.S 89 as 8900 ( as the base is 100) rather than just carry-forward. So, -60 obtained is added to 8900 to get the final answer 8840.

 

Let’s consider another example:

Example : Can you multiply 95 x 101 using this shortcut ?

 

BaseMultiplication8

 

Exercise : On the similar lines, can you mentally multiply?

  • 9×12
  • 95×105
  • 89×104
  • 891×1005

If you have any doubts using this shortcut, please leave a comment. I would be glad to help.

Kiran Chandrashekhar

Hey, Thanks for dropping by. My name is Kiran Chandrashekhar. I am a full-time software freelancer. I love Maths and Mathematical Shortcuts. Numbers fascinate me. I will be posting articles on Mathematical Shortcuts, Software Tips, Programming Tips in this website. I love teaching students preparing for various competitive examinations. Read my complete story.

9 Comments

    • I think, I have already handled in the tutorial. Please write to me, I will send you the material related to this topic

    • @jfluid—
      right hand side 1×5=5 that makes 9595. just an error. thanx for pointing

      • Thanks a lot 🙂 for pointing it out. I have corrected it now.

  1. This info is worth everyone’s attention. Where can I find out
    more?

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