Subtracting Whole Numbers

1. No Borrowing
   
   To subtract whole numbers we write them as in an addition problem and subtract each digit moving from the right to the left.
   
   Example:
   
           789
        -   34        Note that 9 - 4  =  5,    8 - 3  =  5,  and  7 - 0  =  7
           755
   
   

2. Borrowing is Necessary
   
   If when subtracting digits, the top number is smaller than the bottom, borrowing becomes necessary.  We borrow one from the digit to the left.
   
   Example:
   
   Find 41  -  9
   
   Solution
            3
            4 11
        -       9        We have written 41 as 30 + 11.  Notice the three in the new tens digit.
             32
   
   
   Example:
   
   Find 1042  -  578
   
   Solution
             0 9 14            First we write 52 as 40 + 12.  Then notice that 0 can not be borrowed from,
            1 0 512
        -     5 7 8        so we go to the next digit and write 104 as 90 + 14.
               47 4
   

3. Checking Your Work
   
   We can think of subtraction as the reverse of addition.
   
   Example
   
   To check that 41 - 9  =  32, we can work out 32 + 9:
   
            1
            32
        +    9        Since 2 + 9  =  11, we have carried the 1.
            41
   
   Exercise
   
   Check your work on the second example:  1052 - 578  =  474
   

4. Solving for x
   
   We can use this idea that subtraction is the inverse of addition when we answer questions that ask to solve for x:
   
   Example
   
   Solve for x if
   
           15  =  x + 11
   
   Solution
   
   We write the reverse of the addition problem which is a subtraction problem.  We have
   
           x  =  15 - 11 
   
           15
        - 11        Since 2 + 9  =  11, we have carried the 1.
             4
   

5. Applications
   
   Example
    

   Consider the following table taken for the United Nations world population figures 

   (http://www.un.org/esa/population/pubsarchive/pop1999-00/pop1999-00.pdf)
   
   
   
   

   1. How much did the world's total population increase between 1999 and 2000?
      
      

   2. In the year 2000, how many more people were there in less developed countries vs. more developed countries?
      
      

   3. In the year 1999, how many more people were there in more developed countries vs. the least developed countries?

   Solution

    

   1. We subtract the two world populations:
      
               5  9 14 14                           
              6, 0 5 5, 10 4 9, 0 0 0
           -  5, 9 7 8, 4 0 1, 0 0 0        Since the numbers are in thousands, we tag on three 0's
                      7 6, 6 4 8, 0 0 0
      
      Between 1999 and 2000, the world population increased by 76,648,000 poeple.
      

   2. We subtract the more developed number from the less developed number
      
                    15 16  9                           
              4, 8 6 7, 0 16 9, 0 0 0
           -  1, 1 8 7, 9  8 0, 0 0 0        Since the numbers are in thousands, we tag on three 0's
               3, 6 7 9, 0  8 9, 0 0 0
      
      In 2000, there were 3,679,089,000 more people in less developed countries than in more developed countries.
      
      

   3. We subtract the least developed number from the more developed number
      
                 3 14  5                           
              6 4 4, 6 17 7, 0 0 0
           -  6 2 9, 5 8 6, 0 0 0        Since the numbers are in thousands, we tag on three 0's
                 1 5, 0 9 1, 0 0 0
      
      In 1999, there were 15,091,000 more people in more developed countries than in the least developed countries.
      

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