Competition · AMC preparation · step 4 of 4
AMC 10 · 2024A · #1
Grade 7 arithmeticalgebraPick an answer.
AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Brute-force multiplication of 9901 · 101 and 99 · 10101 would give two five-digit numbers and an easy place for a careless slip. Instead, notice that 9901 = 99 · 100 + 1 and 10101 = 101 · 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.
Name the two small numbers
Let a = 99 and b = 101, so the big numbers become 100a + 1 and 100b + 1.
Giving the repeated numbers short names is the Grade 6 "letters stand for numbers" move; it makes the hidden symmetry pop out.
Giving the repeated numbers short names makes the hidden matching pieces pop out.
▸ Why?
Opening each product sends every piece against every piece, so the shared terms line up.
▸ Why?
Subtracting two quantities that hold the identical piece removes it and leaves only the small extras.
Rewrite with the letters
Rewrite the whole expression as (100a + 1) · b - a · (100b + 1), lining the two 100 factors up in parallel.
Same expression, new clothing — the substitution doesn't change the value, but it lines up matching pieces.
6.EE.A.3Change Focus Count The ComplementExpand both products
Distribute both products to get 100ab + b - 100ab - a, where 100ab shows up twice with opposite signs.
Distribute carefully, especially the minus sign across the second parenthesis.
7.EE.A.1Change Focus Count The ComplementCancel and plug back in
The two 100ab terms cancel, leaving b - a = 101 - 99, which is answer (A).
Once the heavy term cancels, only the two tiny extras +b and -a survive — and their difference is just 2.
7.EE.A.1Change Focus Count The ComplementBig-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!
- Name the two small numbers
- Rewrite with the letters
- Expand both products
- Cancel and plug back in
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