Competition · AMC preparation · step 4 of 4
AMC 8 · 2011 · #10
Grade 6 algebraPick an answer.
AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The 10 has to do three different jobs — tip, flat fare for the first 0.5 mile, and the per-0.1-mile charge after that — so Tool #7 (Identify Subproblems) cleanly peels them off one at a time: take out the tip, take out the flat fare, then ask "how far does the leftover money buy?" Tool #8 (Analyze the Units) turns the awkward0.20 per 0.1 mile rate into the friendlier $2 per mile, which makes the final division a one-line decimal computation rather than an algebra problem.
Take out the tip
Set aside the tip first: you hand over the $2 at the end, leaving only $8 for the meter.
Splitting the $10 into "tip" and "fare" is the Tool #7 move — handle one job at a time.
4.MD.A.2Identify SubproblemsPay the flat fare
Pay the flat fare for the first mile: subtract $2.40, leaving $5.60 for the extra distance.
Subtracting decimals to the hundredths place is a Grade 5 arithmetic move.
5.NBT.B.7Identify SubproblemsRewrite the rate per mile
Convert the rate: $0.20 per 0.1 mile is the same as $2 per mile — a friendlier unit.
Scaling the top and bottom of the rate by 10 turns "per 0.1 mile" into "per mile" — a Grade 6 unit-rate move.
6.RP.A.3Analyze The UnitsDivide to get extra miles
At $2 per mile, the leftover $5.60 buys 2.8 extra miles (5.60 ÷ 2).
Dividing 5.60 by 2 is a direct Grade 5 decimal calculation.
After the tip and the flat fare for the first 1/2 mile are paid, the leftover $5.60 stretches to 2.8 more miles of riding.
▸ Why?
The leftover $5.60 buys extra distance at a steady $2.00 for every mile.
▸ Why?
The meter's rate of $0.20 per 0.1 mile is the same price as $2.00 per whole mile, because one mile is ten of those 0.1-mile pieces and each piece costs $0.20.
▸ Why?
One whole mile is exactly ten equal pieces of 0.1 mile.
▸ Why?
Ten pieces that each cost $0.20 cost ten times $0.20 altogether.
▸ Why?
Once each mile costs a fixed $2.00, the number of miles is the leftover money split into $2.00 chunks, one chunk for each mile.
▸ Why?
Finding how many $2.00 chunks fit inside $5.60 is division, which undoes the multiplication that turns a distance in miles into a cost in dollars.
Add back the first half mile
Add the first 0.5 mile back to the 2.8 extra: 0.5 + 2.8 = 3.3 miles — choice (C).
Re-combining the two subproblem pieces (first 0.5 mile + extra 2.8 mile) finishes the Tool #7 split.
5.NBT.B.7Identify SubproblemsBig AMC 8 word problems often shrink down to Grade 6 rate reasoning — once you split off the tip and the flat fare, only a tidy decimal division is left.
- Take out the tip
- Pay the flat fare
- Rewrite the rate per mile
- Divide to get extra miles
- Add back the first half mile
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