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?

the painting is 22.5 inches tall and 30 inches wide. 

when the paper is vertically oriented, it is 11 inches tall and 8.5 inches wide.
take away 1 inch border and the usable dimensions would be 9 inches tall and 8.5 inches wide. 

if you use the height as the controlling ratio, then the scale factor would be 9/22.5. 

22.5 inches tall * 9/22.5 = 9 inches tall.
30 inches wide * 9/22.5 = 12 inches wide. 

the ratio of height to width is preserved since 22.5/30 = .75 and 9/12 = .75. 

however, the 12 inches width is greater than the 6.5 inches width allowed, so this doesn't work. 

if you use the width as the controlling ratio, then the scale factor would be 6.5/30.

22.5 inches tall * 6.5/30 = 4.875 inches tall.
30 inches wide * 6.5/30 = 6.5 inches wide. 

the ratio of height to width is preserved since 22.5/30 = .75 and 4.875/6.5 = .75 

both height and width are now within the outer limits of 9 inches tall and 6.5 inches wide, so these measurements are good and the scale factor is 6.5/30 which is equivalent to 13/60 is you want it in rational number form. 

you would do similar analysis when the paper is horizontally oriented. 

you would find that the height is now the controlling factor and the scale factor would be 6.5/22.5. 

you would get a height on the paper of 22.5 * 6.5/22.5 = 6.5.
you would get a width on the paper of 30 * 6.5/22.5 = 8.67. 

the height to width ratio on the paper would be .75, same as on the painting, and the height and width limits on the paper would be adhered to: 

they are less than or equal to 6.5 inches on the height and less than or equal to 9 inches on the width. 

so, to the best of my knowledge, your solutions would be: 

scale factor of 6.5/30 = 13/60 if the paper is vertically oriented. 

scale factor of 6.5/22.5 = 13/45 is the paper is horizontally oriented.