AMC 10 · 2020 · #4

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

rateunit-conversiondimensional-analysismulti-digit-arithmetic dimensional-analysisidentify-subproblems ↑ Prerequisites: rate

📏 Medium solution 💡 2 insights

Problem

A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $0.50$ per mile, and her only expense is gasoline at$ 2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: A driver goes $60$ mph for $2$ hours in a car that gets $30$ miles per gallon. She is paid $\$0.50$ per mile, and her only cost is gasoline at $\$2.00$ per gallon. What is her net pay rate in dollars per hour?

Givens: Speed: $60$ miles/hour for $2$ hours; Fuel economy: $30$ miles/gallon; Pay: $\$0.50$ per mile; Gas cost: $\$2.00$ per gallon; Answer choices: (A) $20$, (B) $22$, (C) $24$, (D) $25$, (E) $26$

Unknowns: Net dollars per hour after subtracting gas

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities

Tool #8 (Analyze the Units) is perfect — the question is built entirely from rates (mile/hour, mile/gallon, $/mile, $/gallon, $/hour). Track units through and the formula writes itself. Tool #7 (Subproblems) splits the work into earnings rate, gas-cost rate, and net rate — all per hour, so we don't even need the total $120$ miles. Tool #3 (Eliminate) checks the magnitude: gross is $\$30$/hr (since $60$ mi/hr $\times \$0.50$/mi $= \$30$/hr), so the net must be a bit less than $30$ — answers $\le 20$ are out.

Execute — Answer: E

#8 Analyze the Units 6.RP.A.3 Step 1

Gross pay per hour.
Multiply speed (mile/hour) by pay rate ($/mile): the "mile" units cancel and leave $/hour.

$60\ \dfrac{\text{mi}}{\text{hr}} \times \dfrac{\$0.50}{\text{mi}} = \$30\ \text{per hour}$

💡 Speed $\times$ pay-per-mile gives pay-per-hour because miles cancel.

#7 Identify Subproblems 6.RP.A.3 Step 2

Gas cost per hour.
Each hour she covers $60$ miles, which burns $60/30 = 2$ gallons, costing $2 \times \$2.00 = \$4$.

$60\ \dfrac{\text{mi}}{\text{hr}} \times \dfrac{1\ \text{gal}}{30\ \text{mi}} \times \dfrac{\$2.00}{\text{gal}} = \$4\ \text{per hour}$

💡 Chain three rates so miles and gallons both cancel — left with dollars per hour.

#7 Identify Subproblems 4.OA.A.3 Step 3

Net pay per hour is gross minus gas: $\$30 - \$4 = \$26$ per hour.
That's choice (E).

$\$30/\text{hr} - \$4/\text{hr} = \$26/\text{hr} \;\Rightarrow\; \textbf{(E)}$

💡 Earnings minus the only expense leaves net pay.

#3 Eliminate Possibilities 6.RP.A.3 Step 4

Eliminate sanity check: gross is $\$30$/hr, so net has to be less than $30$. Choice (A) $20$ would mean $\$10$/hr in gas, but we computed only $\$4$/hr — too low for (A). (E) $26 = 30 - 4$ matches exactly.

$\text{gross} = \$30/\text{hr}, \quad \text{gas} = \$4/\text{hr} \;\Rightarrow\; \textbf{(E)}$

💡 Net $=$ gross minus expense; only one choice fits the right gap.

[1] #8 6.RP.A.3 Gross pay per hour. Multiply speed (mile/hour) by pay rate ($/mile): the "mile"

[2] #7 6.RP.A.3 Gas cost per hour. Each hour she covers $60$ miles, which burns $60/30 = 2$ gall

[3] #7 4.OA.A.3 Net pay per hour is gross minus gas: $\$30 - \$4 = \$26$ per hour. That's choice

[4] #3 6.RP.A.3 Eliminate sanity check: gross is $\$30$/hr, so net has to be less than $30$. Cho

Review

Reasonableness: Whole-trip check (units in dollars). Distance $= 60 \times 2 = 120$ mi. Earnings $= 120 \times 0.50 = \$60$. Gas used $= 120/30 = 4$ gal, costing $4 \times 2 = \$8$. Net $= 60 - 8 = \$52$ over $2$ hours $= \$26$/hr. Matches (E).

Alternative: Tool #6 (Guess and Check) on the choices. For each choice $r$, see whether $r \times 2$ hours equals $\$52$ net. $r = 26$: $26 \times 2 = \$52\ \checkmark$. No other choice gives $\$52$.

CCSS standards used (min grade 6)

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Subtracting gas cost from gross to get net, and computing the per-hour totals.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Chaining the four rates (mi/hr, $/mi, mi/gal, $/gal) so units cancel and give $/hr.)

⭐ This AMC 10 problem only needs Grade 6 "ratios and rates" you already know — earnings $\$30$/hr, gas $\$4$/hr, net $\$26$/hr.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 6)

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