Let → OA = → a , → OB = 12 → a + 4 → b and → OC = → b , where O is...

Area of parallelogram, $S = | \overset{\to}{a} \times \overset{\to}{b} |$

Area of quadrilateral = Area \left(∆ OAB\right) +$Area \left(∆ OBC\right)$

$= \dfrac{1}{2} { | \overset{\to}{a} \times ( 12\overset{\to}{a} + 4\overset{\to}{b} ) | + | \overset{\to}{b} \times ( 12\overset{\to}{a} + 4\overset{\to}{b} ) | }$

$= \dfrac{1}{2} { | 4\overset{\to}{a} \times \overset{\to}{b} | + | 12\overset{\to}{b} \times \overset{\to}{a} | }$

$= \dfrac{1}{2} { 4 | \overset{\to}{a} \times \overset{\to}{b} | + 12 | \overset{\to}{a} \times \overset{\to}{b} | }$

$= \dfrac{16}{2} | \overset{\to}{a} \times \overset{\to}{b} |$

$= 8 | ( \overset{\to}{a} \times \overset{\to}{b} ) |$

Ratio $= \dfrac{8 | ( \overset{\to}{a} \times \overset{\to}{b} ) |}{| ( \overset{\to}{a} \times \overset{\to}{b} ) |} = 8$