AMC 10 · 2021 · #5
Grade 5 rate-ratioPick an answer.
Every number here is either a distance (feet) or a count of steps, and what the question wants is feet per stride and feet per leap. Watching the units tells you exactly which division to do: feet divided by number of steps gives feet per step. The one place units cannot help is the pole count, so a quick sketch settles how many gaps 41 poles create before any dividing starts.
Count gaps, not poles
Count the gaps, not the poles.
The first pole starts the line without creating a gap behind it, so poles always outnumber gaps by exactly one.
The first pole starts the line without creating a gap behind it, so poles outnumber gaps by exactly one.
▸ Why?
Each gap sits between two neighbouring poles, so gaps and neighbouring pairs match one for one.
▸ Why?
The poles are set at the same fixed spacing, so the whole distance is that step counted once per gap.
Find one gap in feet
Find one gap's length.
Equal spacing means the mile is cut into 40 identical pieces, so one piece is the mile divided by 40.
5.NBT.B.6Identify SubproblemsDivide the same 132 feet two ways
Divide that length two ways.
One fixed distance shared among more steps means shorter steps, so the same 132 feet gives a small stride and a big leap.
5.NBT.B.6Analyze The UnitsSubtract to compare
Subtracting gives 8 feet.
"How much longer" always means the difference of the two lengths, so the last move is a subtraction.
4.NBT.B.4Identify SubproblemsCount the gaps between the poles, not the poles, and then let the units guide you: feet divided by number of steps gives the length of one step.
- Count gaps, not poles
- Find one gap in feet
- Divide the same 132 feet two ways
- Subtract to compare