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Column Multiplication: Columns

Sometimes, you will see an equation that you can't solve just by using simple multiplication. Here's an example that uses two columns:

 
 
 
4
2
×
 
 
3
1

   
 
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
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Right To Left. We first see the 2, which is above the 1.  2 × 1 = 2. Our equation is now:

 
 
 
4
2
×
 
 
3
1

   
 
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
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Because there is a 4 next to the 2, we do 4 × 1 = 4. Our equation is now:

 
 
 
4
2
×
 
 
3
1

   
 
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
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We are finished with the 1 in the second row. So, going from Right To Left we see the 3. But before we start again from Right to Left in the first row, we must write a zero in the rightmost box in the next row where we are writing the results:

 
 
 
4
2
×
 
 
3
1

    
  
    
  
    
  
    
  
   
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
   White Box   White Box   White Box   White Box

Going from Right to Left in the first row, 2 × 3 = 6.  The equation is now:

 
 
 
 
4
2
×
 
 
 
3
1

    
  
    
  
    
  
    
  
   
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
   White Box   White Box   White Box   White Box

Again we go from Right To Left in the second row and see the 3.  4 × 3 = 12.  The equation is now:

 
 
 
 
4
2
×
 
 
 
3
1

    
  
    
  
    
  
    
  
   
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
   White Box   White Box   White Box   White Box

We're almost finished. Now, we must use what we learned in Column Addition to add 42 plus 1260:

 
 
 
 
 
4
2
×
 
 
 
3
1

    
  
    
  
    
  
    
  
   
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
   White Box   White Box   White Box   White Box

Because of column addition, we put a 1 in the leftmost box in the first row. The answer is 1302.
We didn't write a plus sign when we added because most people don't: they know that they have to so addition.

The following example uses three columns:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

Right to Left. We go from right to left in the first row. We start by multiplying each number in the first row by the rightmost number in the second row. 1 × 2 = 2:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

We continue to go from right to left in the first row.  0 × 2 = 0:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

The leftmost number in the first row is 2.  2 × 2 = 4:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

We are finished with the 2 in the second row. So, going from Right To Left in the second row we see the 1. But before we start again from Right to Left in the first row, we must write a zero in the rightmost box in the next row where we write the results. We do this because the 1 is one number away from the 2:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

Right to Left. We go from right to left in the first row. We multilply each number in the first row by the 1 in the second row. 1 × 1 = 1:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

We continue to go from right to left in the first row.  0 × 1 = 0:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

The leftmost number in the first row is 2.  2 × 1 = 1:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

We are finished with the 1 in the second row. So, going from Right To Left in the second row we see the 3. But before we start again from Right to Left in the first row, we must write a zero in the two rightmost boxes in the next row where we write the results. We do this because the 3 is two numbers away from the 2:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

Right to Left. We go from right to left in the first row. We multilply each number in the first row by the 3 in the second row. 1 × 3 = 3:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

We continue to go from right to left in the first row.  0 × 3 = 0:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

The leftmost number in the first row is 2.  2 × 3 = 6:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

We're almost finished. Now, we must use what we learned in Column Addition to add 402 plus 2010 plus 60300:

 
 
 
2
0
1
×
 
 
3
1
2

   
 
 
 
  
 
  
 
  
   
 
 
  
 
  
 
  
 
  
   
 
  
 
  
 
  
 
  
 
  

   
 
  
 
  
 
  
 
  
 
  

The answer is 62712.

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