AMC 10 · 2016 · #7
Grade 8 rate-ratioPick an answer.
AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The ratio 5:4 is the natural place to plant a single variable: write the angles as 5x and 4x, and the whole problem collapses to one unknown (Tool #4). From there the two English conditions — 'complement' and 'twice as large' — turn directly into algebra (Tool #13): each complement is 90 minus the angle, and 'twice as large' is an equals sign with a factor of 2. Identifying which complement is the bigger one is a small but essential subproblem (Tool #7): the larger angle has the smaller complement, so the doubled complement must be the one belonging to the 4x angle. Pin that down and a single linear equation gives x.
Name the angles from the ratio
The ratio 5:4 shares one unit x, so write the larger and smaller angles as 5x and 4x — two unknowns become one.
A ratio is just a recipe of equal parts, so one part size x describes both angles at once.
6.RP.A.3Use Matrix LogicWrite each complement
Each complement is 90 minus the angle: 5x gives 90 - 5x, and the smaller 4x gives the bigger complement 90 - 4x.
Bigger angle eats up more of the 90°, so it leaves a smaller complement behind.
7.G.B.5Convert To AlgebraTurn 'twice as large' into an equation
The bigger complement (from 4x) is twice the smaller (from 5x), so 90 - 4x = 2(90 - 5x).
'Twice as large' is literally an equals sign with a factor of 2 on the smaller quantity.
7.EE.B.4Convert To AlgebraSolve for the part size
Expand and collect: 90 - 4x = 180 - 10x, so 6x = 90 and x = 15.
Gather the unknown on one side and the numbers on the other, then divide to free x.
8.EE.C.7Use Matrix LogicAdd the two angles
With x = 15 the angles are 5x = 75° and 4x = 60°, both acute, so the sum is 135. Answer (C).
Once x is known, the sum is just 9 copies of that part size.
7.EE.B.3Identify SubproblemsTurn a ratio into equal parts named x, then let each plain-English clue become one piece of an equation.
- Name the angles from the ratio
- Write each complement
- Turn 'twice as large' into an equation
- Solve for the part size
- Add the two angles