AMC 10 · 2008 · #9
Grade 8 rate-ratio
Pick an answer.
The ratios 4:3 and 2:1 fix the shapes but not the actual sizes, so Tool #4 (Introduce a Variable) names one scale factor k and writes every length in terms of it. The picture (Tool #1) shows a right triangle made by the width, height, and diagonal, which turns the single number 27 into an equation for k. Then Tool #7 (Identify Subproblems) breaks the goal into a chain: find the screen size, find the movie height, then find the leftover and halve it.
Name the screen size with one variable
One variable builds the ratio straight into the screen.
One scale factor times the ratio numbers rebuilds every actual length while locking the shape in place.
6.RP.A.3Introduce A VariableUse the diagonal to solve for k
The diagonal pins that variable down.
A 4:3 rectangle is just a scaled 3-4-5 triangle, so the diagonal is always 5 of the same units.
A four-by-three rectangle is a scaled three-four-five triangle, so its diagonal is always five of the same units.
▸ Why?
The diagonal, the width and the height form a right triangle, so the squares on the sides add to the square on the diagonal.
▸ Why?
One scale factor stretches every length together, so the shape is locked and only its size is unknown.
Find the screen's width and height
That gives both screen dimensions.
Multiplying the scale factor by each ratio number gives the real measurements.
5.NBT.B.7Introduce A VariableFind the movie's height
The movie fills the width, so its own ratio gives its height.
A 2:1 shape is exactly twice as wide as it is tall, so its height is half its width.
6.RP.A.3Identify SubproblemsTake the leftover and split it in two
Halving the leftover gives 2.7, choice (D).
Whatever height the movie does not use is dark, and the two equal strips each get half of it.
5.NBT.B.7Identify SubproblemsA 4:3 screen is a scaled 3-4-5 triangle, so a 27-inch diagonal makes it 16.2 tall; the 10.8-tall movie leaves 5.4 of darkness split into two 2.7-inch strips.
- Name the screen size with one variable
- Use the diagonal to solve for k
- Find the screen's width and height
- Find the movie's height
- Take the leftover and split it in two