AMC 10 · 2024 · #1
Grade 7 arithmeticalgebraProblem
What is the value of 9901⋅101−99⋅10101?
Pick an answer.
AMC 10 2024 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.
Toolkit + CCSS Solution
Understand
Restated: Compute the value of $9901 \cdot 101 - 99 \cdot 10101$ and match it to one of the five answer choices.
Givens: Expression: $9901 \cdot 101 - 99 \cdot 10101$; Four key numbers: $9901$, $101$, $99$, $10101$; Answer choices: (A) $2$, (B) $20$, (C) $200$, (D) $202$, (E) $2020$
Unknowns: The numerical value of the expression
Understand
Restated: Compute the value of $9901 \cdot 101 - 99 \cdot 10101$ and match it to one of the five answer choices.
Givens: Expression: $9901 \cdot 101 - 99 \cdot 10101$; Four key numbers: $9901$, $101$, $99$, $10101$; Answer choices: (A) $2$, (B) $20$, (C) $200$, (D) $202$, (E) $2020$
Plan
Primary tool: #16 Use Symmetry / Transform
Secondary: #4 Introduce a Variable
Brute-force multiplication of $9901 \cdot 101$ and $99 \cdot 10101$ would give two five-digit numbers and an easy place for a careless slip. Instead, notice that $9901 = 99 \cdot 100 + 1$ and $10101 = 101 \cdot 100 + 1$ — both numbers split into a product of the other given factors plus $1$. That symmetry is exactly what Tool #16 looks for: rewrite the expression so a large piece cancels. Tool #4 (let $a = 99$, $b = 101$) makes the symmetry visible at a glance and turns the whole problem into one line of algebra.
Execute — Answer: A
6.EE.A.2 Step 1 - Name the two small numbers.
- Let $a = 99$ and $b = 101$.
- Now look at the two big numbers and write each one in terms of $a$ and $b$.
💡 Giving the repeated numbers short names is the Grade 6 "letters stand for numbers" move; it makes the hidden symmetry pop out.
6.EE.A.3 Step 2 - Rewrite the original expression using $a$ and $b$.
- Both factors of $100$ now appear in a parallel way.
💡 Same expression, new clothing — the substitution doesn't change the value, but it lines up matching pieces.
7.EE.A.1 Step 3 - Use the distributive property to expand both products.
- The $100ab$ term shows up in each part with opposite signs.
💡 Distribute carefully, especially the minus sign across the second parenthesis.
7.EE.A.1 Step 4 - Cancel the matching $100ab$ terms and plug $a$ and $b$ back in.
- The huge calculation collapses to a single subtraction.
💡 Once the heavy term cancels, only the two tiny extras $+b$ and $-a$ survive — and their difference is just $2$.
6.EE.A.2 Name the two small numbers. Let $a = 99$ and $b = 101$. Now look at the two big 6.EE.A.3 Rewrite the original expression using $a$ and $b$. Both factors of $100$ now app 7.EE.A.1 Use the distributive property to expand both products. The $100ab$ term shows up 7.EE.A.1 Cancel the matching $100ab$ terms and plug $a$ and $b$ back in. The huge calcula Review
Reasonableness: The answer choices jump from $2$ to $20$ to $200$ to $202$ to $2020$, so they're testing whether the student noticed the cancellation. A direct check confirms it: $9901 \cdot 101 = 1{,}000{,}001$ and $99 \cdot 10101 = 999{,}999$, and $1{,}000{,}001 - 999{,}999 = 2$. The two giant products are within $2$ of each other, exactly matching $b - a = 101 - 99$, so $\textbf{(A)}$ is the only consistent choice.
Alternative: Tool #6 (Guess and Check by direct computation): multiply $9901 \cdot 101 = 1{,}000{,}001$ and $99 \cdot 10101 = 999{,}999$, then subtract to get $2$. It works but burns time and invites arithmetic mistakes — the algebraic transform is much safer under contest pressure.
CCSS standards used (min grade 7)
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Letting $a = 99$ and $b = 101$ so the four numerical inputs can be written as $a$, $b$, $100a + 1$, and $100b + 1$.)6.EE.A.3Apply the properties of operations to generate equivalent expressions (Substituting the rewritten forms of $9901$ and $10101$ to turn the original expression into the equivalent expression $(100a + 1)b - a(100b + 1)$.)7.EE.A.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients (Expanding $(100a + 1)b - a(100b + 1)$ with the distributive property, then cancelling the $100ab$ terms to reduce the expression to $b - a$.)
⭐ Big-looking numbers often hide a small structure. Naming $99$ and $101$ as $a$ and $b$ exposes the matching $100ab$ piece that cancels — leaving the AMC 10 opener as a Grade 7 distributive-property exercise!
⭐ Big-looking numbers often hide a small structure. Naming $99$ and $101$ as $a$ and $b$ exposes the matching $100ab$ piece that cancels — leaving the AMC 10 opener as a Grade 7 distributive-property exercise!
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