Competition · AMC preparation · step 4 of 4
AMC 8 · 2009 · #6
Grade 6 rate-ratioPick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a fill-rate problem: time = volume / rate. The rate is given in gallons per minute but the answer must be in hours, so Tool #8 (Analyze the Units) keeps the conversion gal/min → gal/hr honest. Tool #7 (Identify Subproblems) splits the work into clean pieces: (i) combine the 4 hoses into a single flow rate, (ii) divide the total volume by that rate to get minutes, (iii) convert minutes to hours.
Combine the four hoses
All 4 hoses run at once, so add their rates: 4 × 2.5 = 10 gallons per minute.
Adding identical rates together is the Tool #7 subproblems move — treat the 4 hoses as one 10 gal/min super-hose. Multiplying a whole number by a decimal to hundredths is the Grade 5 calculation.
5.NBT.B.7Identify SubproblemsConvert to gallons per hour
An hour has 60 minutes, so 10 gal/min × 60 = 600 gal/hr.
The "minute" units cancel and leave gallons per hour — exactly the Grade 5 standard-unit conversion move.
5.MD.A.1Analyze The UnitsDivide volume by rate
With units matched, apply time = volume ÷ rate: divide the pool's capacity by the hourly rate.
Dividing a total quantity by a unit rate to recover time is Grade 6 rate reasoning.
The time to fill the pool equals the pool's capacity divided by the combined rate at which water flows in.
▸ Why?
Water enters at a steady rate, so the total volume in the pool is that rate multiplied by the time spent filling; to recover the time we undo that multiplication.
▸ Why?
A steady rate means every equal stretch of time pours in the same amount of water, so the total volume is that fixed amount counted once for each stretch — a rate times a time.
▸ Why?
Because the volume equals the rate times the time, the time is found by dividing the volume by the rate, since division reverses multiplication.
This AMC 8 problem only needs Grade 6 rate reasoning — total amount divided by rate gives the time — that you already know!
- Combine the four hoses
- Convert to gallons per hour
- Divide volume by rate
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