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Soln:
Assume - 12w = 3x = 4y = 12
=> w = 1; x = 4; y = 3

Side of square = w+x = 5 = 2w + y

there are four triangles:
Area T1 = 1/2*w*x = 1/2*1*4 = 2
Area T2 = 1/2*w*x = 1/2*1*4 = 2
Area T3 = 1/2*x*y = 1/2*4*3 = 6
Area T4 = 1/2*w*2w = 1/2*1*2*1 = 1

Total area of triangles = AT1 + AT2 + AT3 + AT4 = 11
thus area of shaded region = Area of square - total area of triangles = 5^2 - 11 = 14

thus fractional part of square that is shaded = area of shaded region/area of square = 14/25

Ans: (B)

OA:

Since 12w=3x=4y,

w=3:12=1:4 and x:y=4:3

so, w = 1
x = 4
y = 3

the fractional part of the square is shaded:
{(w+x)^2 - [(1/2)wx + (1/2)wx +(1/2)xy + (1/2)w(2w)]}/(w+x)^2

= {(w+x)^2 - [wx + (1/2)xy + w^2)]}/[(w+x)^2]

=[(5^2) -(4+6+1)]/(5^2)

= (25 - 11)/25

= 14/25

replace x by 4w and y by 3w
now each side is equl to 5w
area of square = 25w^2

now find out area of 4 triangles on corner
w*x/2 + w*x/2 + w*2w/2 + x*y/2
=> w*4w/2 + w*4w/2 + w*2w/2 + 3w*4w/2 = 11w^2

area of gray part = 25w^2 - 11w^2 = 14w^2

ratio = 14/25

(0) Fraction = Shaded Area / Square Area

(1) Square Area: The area of a square is the product of it's sides, thus: Square Area
= (x+w)²

(2) Shaded Area
= Square Area - White Parts

(2a) White Parts
= Sum of the 4 Triangle Areas
= 0.5xw + 0.5xw + 0.5w2w + 0.5xy
= xw + w² + 0.5xy

(2b) Shaded Area
= Square Area - White Parts
= (x+w)² - xw - w² - 0.5xy
= (x²+2xw+w²) - xw - w² - 0.5xy
= x² + xw - 0.5xy

(3) Fraction
= Shaded Area / Square Area
= (x² + xw - 0.5xy) / (x+w)²

(4) 12w = 3x = 4y
==> x = 4w, y = 3w

(5) Fraction
= (x² + xw - 0.5xy) / (x+w)²
= ((4w)² + 4ww - 2w3w) / (4w+w)²
= (16w² + 4w² - 6w²) / (5w)²
= (14w²) / (5w)²
= 14w² / 25w²
= 14/25