AMC 10 · 2024 · #2

Grade 8 algebrarate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

linear-equations-two-varsystems-of-equationsrateformula-substitution convert-to-algebraidentify-subproblems ↑ Prerequisites: linear-equations-one-varmulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T=aL+bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimates it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?

Pick an answer.

(A)

240

(B)

246

(C)

252

(D)

258

(E)

264

AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: A hiking-time model is $T = aL + bG$, where $L$ is trail length in miles, $G$ is altitude gain in feet, and $T$ is time in minutes. Two different trails both take $69$ minutes: one is $1.5$ mi with $800$ ft gain, the other is $1.2$ mi with $1100$ ft gain. Using the same model, find $T$ for a trail that is $4.2$ mi long with $4000$ ft gain.

Givens: Model: $T = aL + bG$ with unknown constants $a, b$; Trail 1: $L=1.5,\ G=800,\ T=69$; Trail 2: $L=1.2,\ G=1100,\ T=69$; Target trail: $L=4.2,\ G=4000$; Answer choices: (A) $240$, (B) $246$, (C) $252$, (D) $258$, (E) $264$

Unknowns: The estimated time $T$ for the $4.2$ mi, $4000$ ft trail

Understand

Plan

Primary tool: #13 Convert to Algebra

Secondary: #7 Identify Subproblems

The problem hands us a formula with two unknown constants and two data points, which is the textbook setup for Tool #13 (Convert to Algebra) — translate each trail into one linear equation, then solve the $2\times 2$ system. Tool #7 (Identify Subproblems) splits the work into two clean parts: (i) recover the constants $a$ and $b$ from Trails 1 and 2, then (ii) plug them into $T = aL + bG$ for the target trail. The two scenarios share the same $T = 69$, so subtracting (or equating) the equations cancels $T$ immediately and gives a one-line relation between $a$ and $b$.

Execute — Answer: B

#13 Convert to Algebra 8.F.B.4 Step 1

Subproblem 1: write a linear equation for each trail.
Plug each $(L, G, T)$ triple into $T = aL + bG$.

$$\begin{cases} 1.5a + 800b = 69 & \text{(Trail 1)} \\ 1.2a + 1100b = 69 & \text{(Trail 2)} \end{cases}$$

💡 Grade 8 "construct a function from data": each trail is one $(L, G) \mapsto T$ data point, and we are pinning down the linear model that fits both.

#13 Convert to Algebra 8.EE.C.8 Step 2

Both equations equal $69$, so the right-hand sides are equal.
Subtract Trail 2 from Trail 1 to eliminate the constant and get a direct relation between $a$ and $b$.

$$1.5a + 800b = 1.2a + 1100b \;\Rightarrow\; 0.3a = 300b \;\Rightarrow\; a = 1000b$$

💡 Grade 8 systems of equations: equal $T$ values let us subtract the two equations and collapse a 2-variable system into a single relation $a = 1000b$.

#13 Convert to Algebra 7.EE.B.4 Step 3

Substitute $a = 1000b$ back into Trail 1 to solve for $b$, then read off $a$.

$$1.5(1000b) + 800b = 69 \;\Rightarrow\; 2300b = 69 \;\Rightarrow\; b = \tfrac{3}{100},\ a = 1000b = 30$$

💡 Grade 7 one-variable equation: once one unknown is written in terms of the other, the system reduces to $2300b = 69$, which divides cleanly.

#7 Identify Subproblems 8.F.B.4 Step 4

Subproblem 2: plug the recovered constants and the new trail's $(L, G) = (4.2,\ 4000)$ into the model.

$$T = 30(4.2) + \tfrac{3}{100}(4000) = 126 + 120 = 246 \;\Rightarrow\; \textbf{(B)}$$

💡 Now that the model is fully known, applying it to a new $(L, G)$ is just one substitution — the second subproblem is a single line of arithmetic.

[1] #13 8.F.B.4 Subproblem 1: write a linear equation for each trail. Plug each $(L, G, T)$ trip

[2] #13 8.EE.C.8 Both equations equal $69$, so the right-hand sides are equal. Subtract Trail 2 f

[3] #13 7.EE.B.4 Substitute $a = 1000b$ back into Trail 1 to solve for $b$, then read off $a$.

[4] #7 8.F.B.4 Subproblem 2: plug the recovered constants and the new trail's $(L, G) = (4.2,\

Review

Reasonableness: Sanity-check the constants on Trail 2: $1.2(30) + 1100(0.03) = 36 + 33 = 69$. Matches. The numbers also pass a units check: $a = 30$ min/mile and $b = 0.03$ min/ft, so a longer, steeper trail naturally takes more time. For the target, $4.2$ mi is about $3\times$ longer than Trail 1 and $4000$ ft is $5\times$ steeper, so a time of $246 \approx 3.5 \times 69$ is in the right ballpark.

Alternative: Tool #3 (Eliminate Possibilities): once $a = 30$ and $b = 3/100$ are known, you could plug them into each answer choice and verify $30(4.2) + 0.03(4000) = 246$ matches only (B). Or, without solving the system, notice $4.2 = 1.5 + 1.2 + 1.5$ and $4000 = 800 + 1100 + ?$ — but that linear-combination shortcut is messier than just solving the system, which is why Tool #13 is the cleaner primary choice.

CCSS standards used (min grade 8)

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations (Setting up and solving the $2\times 2$ system $1.5a + 800b = 69$ and $1.2a + 1100b = 69$ for the constants $a$ and $b$.)
8.F.B.4 Construct a function to model a linear relationship between two quantities (Reading $T = aL + bG$ as a linear model in $L$ and $G$, fitting it to the two data points, then evaluating it at $(L, G) = (4.2,\ 4000)$.)
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities (Reducing the system to the single-variable equation $2300b = 69$ and solving for $b$, then back-substituting to find $a$.)

⭐ Two trails with the same time give two equations in the same two unknowns — solve the little system once, and the model is yours to use on any new trail.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 8)

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