1968 AMC 12 Problem 24

Question

A painting

18''\times24''

is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is:

1:3

1:2

2:3

3:4

1:1

Answer: C

Concepts:rectangle area quadratic ratio and proportion

Difficulty rating: 1690

Small hint:

Let each side strip have width

x

, so each top strip has width

2x

Big hint:

Set the outside area equal to twice

18\cdot24

Solution:

The outside dimensions are

18+2x

and

24+4x.

Since the frame area equals the painting area,

(18+2x)(24+4x)=2(18)(24).

This gives

x^2+15x-54=0,

so

x=3.

The dimensions are

24

and

36,

with ratio

2:3.

Therefore, the correct answer is C.