MSC #7 Completing the Whole – Addition and Subtraction

In MSC #6, we showed that “completing the whole” can greatly simplify addition. Today we will apply this technique in problems involving addition and subtraction.

The type of question in our featured example, “What is the result if the difference between 276 and 149 is deducted from 451?” is often asked in math competitions including MTAP.

This can be expressed as 451 – (276 – 149)

As the wording of the problem suggests, most students would get the difference (276 – 149 = 127) first. This would require one regrouping or “borrowing”.

Then they will deduct 127 from 451 to get the final answer, 324, again with one “borrowing”.

 But this problem can be easily solved by “completing the whole”.

451 – (276 – 149)      =  451 – 276 + 149 (remove the parentheses)

= 451 + 149 – 276 (commutative property of addition)

= (451 + 149) – 276 (associative property of addition)

= 600 – 276

= 324 (using All from 9 and the Last from 10, see MSC #2)

Other examples:

What is the difference between 964 and the sum of 365 and 264?

964 – (365 + 264)   = 964 – 365 – 264

= (964 – 264) – 365

= 700 – 365 = 335

What is 163 deducted from the sum of 763 and 229?

(763 + 229) – 163 = 763 + 229 – 163

                                    = (763 -163) + 229

                                    = 600 + 229

                                    = 829