AMC 10 · 2010 · #1

Grade 7 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
order-of-operationsmulti-digit-arithmeticmental-arithmetic identify-subproblems ↑ Prerequisites: order-of-operations
📏 Short solution 💡 1 insight
Problem
Evaluate the expression 100(100-3)-(100·100-3). In the first part, 100(100-3), the subtraction inside the parentheses is done before the multiplication; in the second part, 100·100-3, the multiplication is done first and 3 is subtracted afterward. What single number does the whole expression equal?

Pick an answer.

(A)
-20,000
(B)
-10,000
(C)
-297
(D)
-6
(E)
0

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The expression is a difference of two separate products. Computing each product on its own turns one messy line into two easy pieces, then a single subtraction finishes it. Naming 100 as a variable also reveals why the two big products nearly cancel.

1STEP 1

Compute the first term

Parentheses first: 100-3=97, so the first term is 100·97=9700.

100(100-3)=100·97=9700
2STEP 2

Compute the second term

No parentheses here, so multiply first: 100·100=10000, then subtract 3 to get 9997.

100·100-3=10000-3=9997
3STEP 3

Subtract the two terms

Subtract: 9700-9997. The second term is bigger, so the difference is negative: -297, choice (C).

9700-9997=-297
Answer
-297
The two products, 9700 and 9997, are very close, so their difference should be small in size, not tens of thousands. That rules out (A) and (B) at a glance, and -297 sits right where two nearly equal numbers should differ.
💡Key takeaway

Break a scary expression into small products first, then the final subtraction is easy.

  • Compute the first term
  • Compute the second term
  • Subtract the two terms

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