2008 AMC 12A Problem 9

Question

Older television screens have an aspect ratio of

4:3.

That is, the ratio of the width to the height is

4:3.

The aspect ratio of many movies is not

4:3,

so they are sometimes shown on a television screen by “letterboxing” — darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of

2:1

and is shown on an older television screen with a

27

-inch diagonal. What is the height, in inches, of each darkened strip?

2

2.25

2.5

2.7

3

Answer: D

Concepts:Pythagorean Triple ratio and proportion

Difficulty rating: 1410

Small hint:

The screen’s width, height, and diagonal are in ratio

4:3:5

Big hint:

The lit movie region keeps the full width but has height equal to half that width

Solution:

Since the sides and diagonal are in ratio

3:4:5,

the height is

\tfrac{3}{5} \cdot 27 = 16.2

inches and the width is

\tfrac{4}{5} \cdot 27 = 21.6

inches.

The movie has aspect ratio

2:1,

so its height is

\tfrac{21.6}{2} = 10.8

inches.

Each darkened strip therefore has height

\dfrac{16.2 - 10.8}{2} = 2.7

inches.

Thus, D is the correct answer.

2008 AMC 12A Problem 9

Answer: D

Difficulty rating: 1410

Small hint:

Big hint:

Solution:

Problem 9 in other years