AMC 10 · 2021 · #1
Grade 6 arithmeticPick an answer.
The direct route is to square the difference and then long-divide by 169, which is slow and error-prone by hand. Tool #15 (Organize Information in More Ways) says to look at 169 and at the difference as products rather than as lone numbers, because 169 is a familiar perfect square. Tool #7 (Identify Subproblems) splits the expression into its three stacked operations so the parentheses get handled first, and Tool #9 (Solve an Easier Related Problem) trades one four-digit division for two small ones once the shared factor is visible.
Do the parentheses first
Do the parentheses first.
Parentheses act like a fence: everything inside collapses to one number before anything outside touches it.
5.OA.A.1Identify SubproblemsSee 169 and 91 as products
Both numbers share thirteen.
An awkward-looking number is often two friendlier numbers in disguise, so check for a shared factor before multiplying anything out.
An awkward-looking number is often two friendlier numbers in disguise, so look for a shared factor first.
▸ Why?
Every number has exactly one prime recipe, so the factors are there to be found rather than invented.
▸ Why?
A shared factor above and below cancels, so spotting it beats multiplying everything out.
Split one hard division into two easy ones
Split the division into two.
Two 13s on the bottom and two 91s on the top means one four-digit division becomes two two-digit ones.
5.NF.B.4Solve An Easier Related ProblemMultiply the leftovers
Multiplying the leftovers gives 49.
Once the 13s cancel, only the 7 hidden inside 91 survives, and there are two of them.
6.EE.A.1Identify SubproblemsBefore multiplying big numbers out, hunt for a shared factor: 169 = 13 × 13 and 91 = 7 × 13, so the whole fraction collapses into 7 × 7.
- Do the parentheses first
- See 169 and 91 as products
- Split one hard division into two easy ones
- Multiply the leftovers