Competition · AMC preparation · step 4 of 4

AMC 10 · 2024A · #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 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.

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.

How to solve

Strategy Convert to Algebra

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 × 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.

1STEP 1

Write an equation per trail

Subproblem 1: plug each trail's (L, G, T) into T = aL + bG to get one linear equation per trail.

1.5a + 800b = 69 & (Trail 1) ; 1.2a + 1100b = 69 & (Trail 2)

2STEP 2

Set the two sides equal

Both equations equal the same time, so subtract them; the constant cancels, leaving a = 1000b.

1.5a + 800b = 1.2a + 1100b → 0.3a = 300b → a = 1000b

3STEP 3

Substitute to find the constants

Substitute a = 1000b into Trail 1: 2300b = 69, giving b = $\frac{3}{100}$ and a = 30.

1.5(1000b) + 800b = 69 → 2300b = 69 → b = 3/100, a = 1000b = 30

4STEP 4

Plug in the new trail

Subproblem 2: plug the constants and the new trail's (4.2, 4000) into the model to get T = 246.

T = 30(4.2) + 3/100(4000) = 126 + 120 = 246 → (B)

Answer

246

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 × longer than Trail 1 and 4000 ft is 5 × steeper, so a time of 246 ≈ 3.5 × 69 is in the right ballpark.

💡Key takeaway

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.

Write an equation per trail
Set the two sides equal
Substitute to find the constants
Plug in the new trail

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Write an equation per trail

Set the two sides equal

Substitute to find the constants

Plug in the new trail

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