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.3Introduce A VariableWrite 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.
A bigger angle eats up more of the right angle, so it leaves a smaller complement behind.
▸ Why?
An angle and its complement add to a fixed right angle, so they are two parts of one whole.
▸ Why?
With that sum held fixed, raising one part must lower the other.
Turn '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.7Introduce A VariableAdd 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