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2024 AMC 10A Problem 2

2024 AMC 10A Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

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?

A)
$240$
B)
$246$
C)
$252$
D)
$258$
E)
$264$

Answer

Key insight

Subtracting the two 69-minute equations gives a = 1000b, so b = 0.03 and a = 30; then evaluate 4.2a + 4000b.

Solution

The two hikes give
$1.5a + 800b = 69, \qquad 1.2a + 1100b = 69.$
Subtract the second from the first: $0.3a - 300b = 0$ , so $a = 1000b$ .

Put this into the first equation: $1500b + 800b = 2300b = 69$ , hence $b = 0.03$ and $a = 30$ .

For the new trail,
$T = 4.2\cdot 30 + 4000\cdot 0.03 = 126 + 120 = 246.$

The answer is $\boxed{\textbf{(B)}\ 246}$ .

Why this works

Two data points determine the two unknown coefficients of a linear model; because both times are equal, subtraction kills the constant and gives a clean relation between $a$ and $b$ . Once the model is known, the question is a single substitution.

Alternative approach

Rescale to avoid decimals: let $a = 10x$ and $b = y/100$ . The equations become $15x + 8y = 69$ and $12x + 11y = 69$ ; since $15 + 8 = 12 + 11 = 23$ and $69 = 3\cdot 23$ , we get $x = y = 3$ . The answer is $42x + 40y = 82\cdot 3 = 246$ .

The trap

Decimal slips when subtracting (0.3a = 300b), or stopping after finding a and b without computing the requested time.

Common mistakes

Decimal slips when subtracting (0.3a = 300b), or stopping after finding a and b without computing the requested time.
Mixing units, e.g. putting $4000$ feet into the miles slot.

Techniques

Set up the equation/formula and compute; no special trick needed