AMC 10 · 2010 · #1

Grade 7 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
order-of-operationsmulti-digit-arithmeticmental-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: order-of-operations
📏 Medium solution 💡 2 insights
Problem
The same three numbers appear in both halves of an expression, nested differently. Find its value.

Pick an answer.

(A)
-4020
(B)
0
(C)
40
(D)
401
(E)
4020
How to solve
Strategy Introduce a Variable

Grinding out 2010-201 and 201-20 works, but it hides why the answer is a small number. The three numbers repeat, so I name them a, b, c and rewrite the expression with no parentheses at all. Then I can see the sign each number carries. The one thing that has to be right is what happens to a parenthesis sitting behind a minus sign, so I settle that as a separate subproblem first.

1STEP 1

Name the repeated numbers

Naming the repeated numbers reveals the shape.

a=20, b=2010, c=201 ⟹ (a-(b-c))+(b-(c-a))
2STEP 2

Settle the minus-parenthesis rule

A minus flips every sign inside its bracket.

(x-y+z)+(y-z)=x+(y-y)+(z-z)=x ⟹ x-(y-z)=x-y+z
3STEP 3

Clear both parentheses

Clearing both brackets lines the terms up.

(a-(b-c))+(b-(c-a))=(a-b+c)+(b-c+a)
4STEP 4

Watch b and c cancel

Two of the three letters cancel entirely.

(a-b+c)+(b-c+a)=(a+a)+(-b+b)+(c-c)=2a
5STEP 5

Put the number back

Putting the number back gives 40, choice (A).

2a=2 × 20=40
Answer
40
Check the claim that b and c are irrelevant by swapping in small numbers: with b=7 and c=3, the expression reads (20-(7-3))+(7-(3-20))=(20-4)+(7+17)=16+24=40, still 2 × 20. The two decoy answers now make sense. Choice (B) 0 is what you get if you assume the mirrored look means everything cancels, but 20 sits outside the first parentheses and inside the second, so both of its appearances end up positive. Choices ± 4020 come from letting 2010 survive instead of 20, which happens if the minus in front of (2010-201) is not distributed.
💡Key takeaway

A minus sign in front of a parenthesis flips the sign of everything inside it, and once the signs are right, 2010 and 201 cancel and only the two 20s are left.

  • Name the repeated numbers
  • Settle the minus-parenthesis rule
  • Clear both parentheses
  • Watch b and c cancel
  • Put the number back

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