2020 AMC 10B Problem 1

P1-10~1 minPrint

Algebraic Manipulation Set up the equation/formula and compute; no special trick needed

What is the value of $1 - (-2) - 3 - (-4) - 5 - (-6)?$

Show solution without answering

Key insight

Taking away a negative number is the same as adding it, so the expression becomes 1 + 2 - 3 + 4 - 5 + 6.

Solution

1. First clean up the signs. Subtracting a negative number is the same as adding the positive number. For example, $5 - (-2) = 7$ , the same as $5 + 2$ .
2. There are three "minus a negative" pieces here:
- $-(-2)$ becomes $+2$ ,
- $-(-4)$ becomes $+4$ ,
- $-(-6)$ becomes $+6$ .
3. The expression is now
$1 + 2 - 3 + 4 - 5 + 6.$
4. Add it in small groups so nothing slips:
$(1 + 2) + (-3 + 4) + (-5 + 6) = 3 + 1 + 1 = 5.$

The answer is $\boxed{\textbf{(D)}\ 5}$ .

Why this works

The only idea here is the sign rule: minus a negative is plus. Rewrite all the signs first, then add in short groups. On the first problem of the contest, being careful matters much more than being fast.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Each "minus a negative" becomes a plus:
$1 - (-2) - 3 - (-4) - 5 - (-6) = 1 + 2 - 3 + 4 - 5 + 6.$
Group the terms in convenient pairs: $(1+2) + (-3+4) + (-5+6) = 3 + 1 + 1 = 5$ .

The answer is $\boxed{\textbf{(D)}\ 5}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 11: Special Factorizations (§11.2, §11.3, §11.5)
Introduction to Algebra, Chapter 22: Special Manipulations