AMC 8 · 2009 · #6

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

rateunit-conversionmulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticunit-conversion

📏 Short solution 💡 3 insights

Problem

Steve's empty pool holds 24,000 gallons when full. Four hoses fill it together, each delivering 2.5 gallons per minute. How many hours does it take to fill the pool?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

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.

1STEP 1

All 4 hoses run at once, so add their rates: 4 × 2.5 = 10 gallons per minute.

4 × 2.5 = 10 gal/min

2STEP 2

An hour has 60 minutes, so 10 gal/min × 60 = 600 gal/hr.

10 gal/min × 60 min/hr = 600 gal/hr

3STEP 3

With units matched, apply time = volume ÷ rate: divide the pool's capacity by the hourly rate.

time = (24,000 gal)/(600 gal/hr) = 40 hr → (A)

Answer

Sanity-check by going through minutes instead of hours: 24,000 ÷ 10 = 2,400 minutes, and 2,400 ÷ 60 = 40 hours — same answer by a different route. Also, 40 hours is a reasonable real-world number for filling a 24,000-gallon pool with only four garden hoses (a bit under two days), so the magnitude is sensible.

💡Key takeaway

This AMC 8 problem only needs Grade 6 rate reasoning — total amount divided by rate gives the time — that you already know!

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