AMC 8 · 2009 · #6

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

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

📏 Short solution 💡 3 insights

Problem

Steve's empty swimming pool will hold $24,000$ gallons of water when full. It will be filled by $4$ hoses, each of which supplies $2.5$ gallons of water per minute. How many hours will it take to fill Steve's pool?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Pool capacity = $24{,}000$ gallons; Number of hoses = $4$; Each hose flow rate = $2.5$ gallons per minute; Answer choices (hours): (A) $40$, (B) $42$, (C) $44$, (D) $46$, (E) $48$

Unknowns: The total time, in hours, to fill the pool

Understand

Restated: 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?

Givens: Pool capacity = $24{,}000$ gallons; Number of hoses = $4$; Each hose flow rate = $2.5$ gallons per minute; Answer choices (hours): (A) $40$, (B) $42$, (C) $44$, (D) $46$, (E) $48$

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems

This is a fill-rate problem: $\text{time} = \text{volume} / \text{rate}$. 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 $\to$ 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.

Execute — Answer: A

#7 Identify Subproblems 5.NBT.B.7 Step 1

Combine the four hoses into one effective flow rate.
Each hose adds $2.5$ gallons every minute, and all $4$ run at the same time, so the combined rate is $4 \times 2.5 = 10$ gallons per minute.

$$4 \times 2.5 = 10 \text{ gal/min}$$

💡 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.

#8 Analyze the Units 5.MD.A.1 Step 2

Convert the rate from gallons per minute to gallons per hour so the final answer comes out in hours.
There are $60$ minutes in an hour, so $10$ gal/min $\times \, 60$ min/hr $= 600$ gal/hr.

$$10 \tfrac{\text{gal}}{\text{min}} \times 60 \tfrac{\text{min}}{\text{hr}} = 600 \tfrac{\text{gal}}{\text{hr}}$$

💡 The "minute" units cancel and leave gallons per hour — exactly the Grade 5 standard-unit conversion move.

#8 Analyze the Units 6.RP.A.3 Step 3

Apply the fill-rate relationship $\text{time} = \dfrac{\text{volume}}{\text{rate}}$ with consistent units (gallons and hours).
Divide the pool's capacity by the combined hourly rate.

$$\text{time} = \dfrac{24{,}000 \text{ gal}}{600 \tfrac{\text{gal}}{\text{hr}}} = 40 \text{ hr} \;\Rightarrow\; \textbf{(A)}$$

💡 Dividing a total quantity by a unit rate to recover time is Grade 6 rate reasoning.

[1] #7 5.NBT.B.7 Combine the four hoses into one effective flow rate. Each hose adds $2.5$ gallon

[2] #8 5.MD.A.1 Convert the rate from gallons per minute to gallons per hour so the final answer

[3] #8 6.RP.A.3 Apply the fill-rate relationship $\text{time} = \dfrac{\text{volume}}{\text{rate

Review

Reasonableness: Sanity-check by going through minutes instead of hours: $24{,}000 \div 10 = 2{,}400$ minutes, and $2{,}400 \div 60 = 40$ 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.

Alternative: Tool #6 (Guess and Check) on the choices: at the combined rate of $600$ gal/hr, each candidate hour count gives a fill amount of (hours) $\times \, 600$. Choice (A) $40 \times 600 = 24{,}000$ matches exactly; (B) $42 \times 600 = 25{,}200$ overshoots; (C), (D), (E) overshoot by even more. Only (A) hits the pool's capacity on the nose.

CCSS standards used (min grade 6)

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Combining the four hoses into a single rate via $4 \times 2.5 = 10$ gallons per minute.)
5.MD.A.1 Convert among different-sized standard measurement units within a given system (Converting the combined rate from $10$ gal/min to $600$ gal/hr so the final time can be expressed in hours.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Dividing total volume by the combined hourly rate, $24{,}000 / 600 = 40$ hours, to get the fill time as a unit-rate calculation.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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