AMC 10 · 2009 · #6
Grade 5 algebraPick an answer.
The question is 'how many different values', so the safe route is to list every legal grouping and evaluate it (Tool #2). Four numbers in a row have exactly five full parenthesizations, a small enough set to write out completely with no guessing. Each grouping is then a short arithmetic subproblem to evaluate (Tool #7). Finally, because two groupings can land on the same number, Tool #3 (Eliminate Possibilities) is used to drop the duplicates so only distinct values remain.
See what parentheses control
Parentheses only change the order of the operations.
Parentheses don't move the numbers; they only choose which operation goes first, so listing the groupings lists every possible outcome.
Parentheses never move the numbers; they only choose which operation goes first.
▸ Why?
Regrouping factors within a product changes nothing, so only a mixed operation can make the grouping matter.
▸ Why?
A multiplication placed across a sum reaches every term, so which side the sum sits on changes the result.
Write out the five groupings
There are exactly five legal groupings.
Recording each grouping as its own expression turns a fuzzy 'try parentheses' into a fixed checklist of five things to evaluate.
5.OA.A.2Make A Systematic ListEvaluate each grouping
Evaluating each gives its own number.
Doing the inside of the parentheses first is exactly what the parentheses are telling you to do.
5.OA.A.1Identify SubproblemsCount the distinct values
Two of them repeat, so the count is 4, choice (C).
'Different values' means each number counts once, so equal results must be merged before counting.
5.OA.A.1Eliminate PossibilitiesParentheses only change which operation goes first, so list every grouping, work each one out, and count the answers that are actually different.
- See what parentheses control
- Write out the five groupings
- Evaluate each grouping
- Count the distinct values