AMC 10 · 2020 · #1

Grade 7 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

multi-digit-arithmeticorder-of-operations pattern-recognitionidentify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 1 insight

Problem

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

Pick an answer.

(A)

-20

(B)

-3

(C)

(D)

(E)

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: Compute the value of $1-(-2)-3-(-4)-5-(-6)$.

Givens: Six numbers strung together with subtraction signs; Three of the numbers are negative: $-2$, $-4$, $-6$; Three of the numbers are positive: $1$, $3$, $5$; Answer choices: (A) $-20$, (B) $-3$, (C) $3$, (D) $5$, (E) $21$

Unknowns: The single number this expression equals

Understand

Restated: Compute the value of $1-(-2)-3-(-4)-5-(-6)$.

Plan

Primary tool: #5 Look for a Pattern

Secondary: #15 Reorganize Information

Tool #5 (Look for a Pattern): the signs alternate in a clean rhythm — every other term is “minus a negative,” which flips to “plus.” Spotting that pattern turns the scary-looking string into a simple add/subtract problem. Tool #15 (Reorganize) helps once the signs are fixed: group the positives and negatives separately so the arithmetic is just two small sums.

Execute — Answer: D

#5 Look for a Pattern 7.NS.A.1 Step 1

Rewrite every “minus a negative” as “plus the positive.” So $-(-2)$ becomes $+2$, $-(-4)$ becomes $+4$, and $-(-6)$ becomes $+6$.

$$1-(-2)-3-(-4)-5-(-6) = 1+2-3+4-5+6$$

💡 Two minus signs back-to-back cancel each other — the same rule we use for $5-(-3)=8$.

#15 Reorganize Information 3.OA.B.5 Step 2

Reorganize: collect the positive numbers and the negative numbers into two separate groups so each sum is easy.

$$(1+2+4+6) + (-3-5) = 13 + (-8)$$

💡 Addition can be reordered (commutative property), so we group the friendly numbers together.

#5 Look for a Pattern 7.NS.A.1 Step 3

Add the positives: $1+2+4+6 = 13$.
Add the negatives: $-3 + (-5) = -8$.
Combine: $13 + (-8) = 5$.

$$13 + (-8) = 5 \;\Rightarrow\; \textbf{(D)}$$

💡 Once the signs are fixed, this is single-digit arithmetic.

[1] #5 7.NS.A.1 Rewrite every “minus a negative” as “plus the positive.” So $-(-2)$ becomes $+2$

[2] #15 3.OA.B.5 Reorganize: collect the positive numbers and the negative numbers into two separ

[3] #5 7.NS.A.1 Add the positives: $1+2+4+6 = 13$. Add the negatives: $-3 + (-5) = -8$. Combine:

Review

Reasonableness: Recompute left-to-right without grouping: $1-(-2)=3$; $3-3=0$; $0-(-4)=4$; $4-5=-1$; $-1-(-6)=5$. Same answer, so (D) is correct.

Alternative: Tool #1 (Draw a Diagram) — plot each step on a number line, jumping right for “minus a negative” and left for plain subtraction. The final landing spot is $5$.

CCSS standards used (min grade 7)

3.OA.B.5 Apply properties of operations as strategies to multiply and divide (Using the commutative property of addition to regroup positives and negatives before summing.)
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers (Rewriting $-(-n)$ as $+n$ and adding signed numbers to reach $5$.)

⭐ This AMC 10 problem only needs Grade 7 “subtracting a negative is adding a positive” you already know — the messy string is really $1+2-3+4-5+6 = 5$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 7)

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