AMC 10 · 2024 · #2
Grade 8 algebrarate-ratioPick an answer.
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.
Write both trails
The shared 69 minutes gives two equations.
Grade 8 "construct a function from data": each trail is one (L, G) ↦ T data point, and we are pinning down the linear model that fits both.
8.F.B.4Convert To AlgebraSet them equal
It collapses to a equals a thousand b.
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.
Two equal outputs let one equation be subtracted from the other, collapsing a two-variable system.
▸ Why?
Subtracting one true equation from another keeps the result true, so the move is legitimate.
▸ Why?
Whatever the two equations share cancels, so only the difference between the two data points survives.
Solve for a and b
Substituting back gives b as three hundredths and a as 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.EE.B.4Convert To AlgebraApply to the new trail
126 plus 120 is 246 minutes.
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.
8.F.B.4Identify SubproblemsTwo 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.