find the width of a picture frame

Question

A picture frame measures 12 cm by 20cm (uniform width). The picture without the frame measures 84 cm ^2. Find the width of the frame 

Answer 1

Let X be the width of the frame 

 

The frame is on all 4 sides (top and bottom, left and right) so the picture is 

 

(12 - 2X) * (20 - 2X) = 84

240 - 40X - 24X + 4X^2 = 84

4X^2 - 64X + 156 = 0

X^2 - 16X + 39 = 0

(X - 13) * (X - 3) = 0

 

X = 13  or  X = 3

 

13 will result in a negative picture size so choose X = 3 cm as the width of the frame

 

Check:  Picture is 6 by 14 = 84 cm^2