Question

You have a painting that is 30 in. wide and 22.5 in. tall. You would like to reproduce it on a sheet of paper that measures 8 1/2 in. by 11 in., leaving at least a 1­in. margin on all four sides. a) What scale should you use if you keep the sheet of paper in the normal upright orientation? Assume that the reproduction will be as large as possible. b) What scale should you use if you turn the paper on its side?

Answer 1

A 1-inch margin on all four sides of an 8 1/2 in. by 11 in. piece of paper leaves a 6 1/2 in. by 9 in. piece of paper to work with.

a) The painting is in landscape orientation, 30 in. wide by 22.5 in. tall.
The usable paper is in portrait orientation, 6 1/2 in. wide by 9 in. tall.

First, let's assume that the 22.5 in. measurement will be shrunken to exactly 9 in. and, keeping the aspect ratio constant, the 30 in. measurement must then be shrunken to some unknown length, x. We then have:
x/30 = 9/22.5
x = 12
Hmm...12 is too large, because we have only 6.5 in. to work with in that direction.

So, assume instead that the 30 in. measurement will be shrunken to exactly 6.5 in. and, keeping the aspect ratio constant, the 22.5 in. measurement must then be shrunken to some unknown length, y. We then have:
y/22.5 = 6.5/30
y = 39/8 or 4 7/8
This will fit on the paper, because we have 9 in. to work with in that direction.
So, a reduction scale of 6.5/30 works best, properly written as 13/60.

b) The painting is in landscape orientation, 30 in. wide by 22.5 in. tall.
The usable paper is also in landscape orientation, 9 in. wide by 6.5 in. tall.

Use a similar strategy to work out the best scale to use for this situation.

Answer 2

Thank you, Candlelight.