AMC 10 · 2016 · #3
Grade 7 arithmeticPick an answer.
The expression looks tangled because three absolute-value bars are nested inside one another, but each bar is just a grouping symbol. Tool #7 (Identify Subproblems) lets us peel the bars one at a time, innermost first, so the scary stack becomes a short list of easy subtractions. Tool #4 (Introduce a Variable) is already half-done for us — x stands for -2016, and substituting that value turns every bar into plain arithmetic. Tool #3 (Eliminate Possibilities) then matches the final number to the one answer choice that fits.
Read the bars as grouping
The bars are read as grouping symbols.
Nested bars are nested boxes — you can only open the next one after the one inside it is done.
5.OA.A.1Identify SubproblemsSubstitute the value of x
Substituting makes the value concrete.
Once the letter becomes a number, the whole expression is just arithmetic you can do by hand.
6.EE.A.2Introduce A VariableInnermost bar: |x| - x
The innermost layer gives 4032.
Taking a positive number minus its own negative copy just doubles it.
Taking a positive number minus its own negative copy just doubles it.
▸ Why?
Subtracting a negative is adding its opposite, so the two copies stack instead of cancelling.
▸ Why?
The minus sign in front reaches every term inside, which is what turns the subtraction into an addition.
Middle bar: subtract |x|
The middle layer gives 2016.
Absolute value only ever changes a negative; a positive passes through untouched.
6.NS.C.7Identify SubproblemsOuter step: subtract x
The last step gives 4032, choice (D).
The very last "-x" sneaks in one more doubling, pushing the total to twice 2016.
7.NS.A.1Eliminate PossibilitiesAbsolute-value bars are just grouping boxes — open them from the inside out, and remember that subtracting a negative means adding.
- Read the bars as grouping
- Substitute the value of x
- Innermost bar: |x| - x
- Middle bar: subtract |x|
- Outer step: subtract x