AMC 10 · 2012 · #6
Grade 7 logicalgebraPick an answer.
The problem gives no numbers, so the safe move is Tool #4 (Introduce a Variable): call the small rounding amount k and write the two rounded numbers as x+k and y-k. Tool #13 (Convert to Algebra) then turns “rounded x minus rounded y” into a single expression that can be simplified and compared to x-y. Tool #3 (Eliminate Possibilities) reads that comparison against the five statements and keeps the one that must hold for every allowed x and y.
Name the rounding amount
One letter names the rounding amount.
When a quantity is described but not numbered, give it a letter so you can compute with it.
6.EE.B.6Introduce A VariableWrite and simplify her estimate
Simplifying shows both changes add.
Subtracting a group that was made smaller adds that shrinkage back, so both roundings pile onto the difference.
Subtracting a group that was made smaller adds that shrinkage back, so both roundings pile onto the difference.
▸ Why?
Shifting the subtracted quantity down by an amount shifts the difference up by exactly that amount.
▸ Why?
Adding a positive amount always lands to the right on the number line, so the estimate exceeds the truth.
Compare to the true value and choose
So the estimate is always larger, choice (A).
A number plus a positive amount sits to the right of it on the number line, so it is larger.
6.NS.C.7Eliminate PossibilitiesMaking the front number bigger and the number you subtract smaller both push a difference up, so this estimate is always too high.
- Name the rounding amount
- Write and simplify her estimate
- Compare to the true value and choose