AMC 10 · 2009 · #1
Grade 4 arithmeticPick an answer.
Naming the number of bagels with a single letter (Tool #4) lets the muffin count and the whole total be written from that one unknown, so the whole-dollar rule becomes one clean condition to check. Working in cents (Tool #8) is the key move: 'a whole number of dollars' just means the cent total ends in 00, i.e. is a multiple of 100. With only six possible bagel counts, Tool #3 (Eliminate Possibilities) then finishes by testing which count actually lands on a multiple of 100.
Work in cents
Working in the small unit keeps everything whole.
Counting in cents turns 'whole number of dollars' into the simple test 'ends in 00.'
4.MD.A.2Analyze The UnitsName the bagels, build the total
Naming one count builds the total with one letter.
One unknown, b, captures the whole week because the muffins are just whatever is left of the five days.
4.OA.A.3Introduce A VariableSet the whole-dollar condition
The whole-unit demand becomes a remainder condition.
A whole-dollar total is just a multiple of 100, so we hunt for the multiple of 100 inside the reachable range.
A whole-dollar total is just a multiple of a hundred cents, so the hunt is for a multiple of a hundred.
▸ Why?
A dollar is a bundle of a hundred cents, so dollars are the higher place of the same measurement.
▸ Why?
A total lands on a whole number of dollars exactly when dividing by a hundred leaves nothing over.
Test the six counts
Testing the six counts leaves 2, choice (B).
With just six cases, checking each is faster than any clever trick and leaves no doubt.
4.OA.B.4Eliminate PossibilitiesCount the money in cents: a whole number of dollars just means the total ends in 00, so find which choice lands there.
- Work in cents
- Name the bagels, build the total
- Set the whole-dollar condition
- Test the six counts