Chapter 3.3, Problem 13P

The standard subtraction algorithm described in the text is ot the only correct subtraction algorithm. The next subtraction algorithm is called adding the complement. For a 3-digit whole number, N, the complemnet of N is 999-N. For example, the complement of 486 is

     $999-486=513$

Notice that regrouping is never needed to calculate the complement of a number. To use the adding-the-complement algorithm to subtract a 3-digit whole number, start by adding the complement of N rather than subtracting N. For example, to solve

     $723-486$

first add the complement of 486:

     $\begin{array}{l}723\\ \frac{+486}{1236}\end{array}$

Then cross out the 1 in the thousands column, and add 1 to the resulting number:

    $\cancel{1}236\to 236+1=237$

Therefore, according to the adding-the- complement algorithm, $723-486=237.$

1. Use the adding-the-complement algorithm to calculate $301-189$ and $295-178$ . Verify that you get the correct answer.
2. Explain why the adding-the-complement algorithm gives you the correct answer to any 3-digit subtraction problem. Focus on the relationship between the original problem and the addition problem in adding the complement. For example, how are the problems $723-486$ and $723+513$ related? Work with the complement relationship, $513=999-486,$ and notice that $999=1000-1.$
3. What are some advantages and disadvantages of the adding-the-complement subtraction algorithm compared with the common subtraction algorithm described in the text?