# 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**