AMC 10 · 2016 · #4
Grade 8 rate-ratioPick an answer.
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
One letter names both angles.
A ratio is just a recipe of equal parts, so one part size x describes both angles at once.
A ratio is a recipe of equal parts, so one part size describes both angles at once.
▸ Why?
Each part is one equal share of the whole, so counting parts is enough to name each angle.
▸ Why?
Both angles are built from the same part, so knowing one angle sizes the part and then the other.
Write each complement
Each complement follows directly.
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 angle has the smaller complement.
'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
Solving gives the part size 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
Adding gives 135, choice (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