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

What is $100(100-3)-(100\cdot100-3)$ ?

Pick an answer.

(A)

-20,000

(B)

-10,000

(C)

-297

(D)

-6

(E)

AMC 10 2010 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.

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Toolkit + CCSS Solution

Understand

Restated: Evaluate the number $100(100-3)-(100\cdot100-3)$. Two products are formed from the same digits, and the second is subtracted from the first.

Givens: First term: $100(100-3)$, which multiplies 100 by the result inside the parentheses.; Second term: $100\cdot100-3$, which multiplies first and then subtracts 3.; The two terms are combined by subtraction.

Unknowns: The single numerical value of the whole expression.

Understand

Restated: Evaluate the number $100(100-3)-(100\cdot100-3)$. Two products are formed from the same digits, and the second is subtracted from the first.

Plan

Primary tool: #7 Identify Subproblems

Secondary: #4 Introduce a Variable

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.

Execute — Answer: C

#7 Identify Subproblems 5.OA.A.1 Step 1

Compute the first term

Handle the parentheses first: $100-3=97$.
Then multiply: $100\cdot97=9700$.

$$100(100-3)=100\cdot97=9700$$

💡 Parentheses tell you what to build before you multiply.

#7 Identify Subproblems 5.NBT.B.5 Step 2

Compute the second term

Here the multiplication comes first because there are no parentheses around the subtraction: $100\cdot100=10000$, then $10000-3=9997$.

$$100\cdot100-3=10000-3=9997$$

💡 Without parentheses, multiply before you subtract.

#7 Identify Subproblems 7.NS.A.1 Step 3

Subtract the two terms

Now subtract the second term from the first: $9700-9997$.
Since 9997 is larger than 9700, the result is negative: $9700-9997=-297$.
That matches choice (C).

$$9700-9997=-297$$

💡 Taking away more than you have lands you below zero.

[1] #7 5.OA.A.1 Handle the parentheses first: $100-3=97$. Then multiply: $100\cdot97=9700$.

[2] #7 5.NBT.B.5 Here the multiplication comes first because there are no parentheses around the

[3] #7 7.NS.A.1 Now subtract the second term from the first: $9700-9997$. Since 9997 is larger t

Review

Reasonableness: 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.

Alternative: Let $n=100$. Then $n(n-3)-(n\cdot n-3)=n^2-3n-n^2+3=-3n+3$. Plugging in $n=100$ gives $-300+3=-297$, confirming (C) without any large multiplication.

CCSS standards used (min grade 7)

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate (Deciding to evaluate inside the parentheses before multiplying in the first term.)
5.NBT.B.5 Fluently multiply multi-digit whole numbers (Computing $100\cdot97$ and $100\cdot100$.)
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers (Subtracting a larger number from a smaller one to get the negative result $-297$.)

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

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Compute the first term

Compute the second term

Subtract the two terms

Review

CCSS standards used (min grade 7)

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