Repeated identity question.

Ratio of Triangle area to Rectangle area = 4 : 9

Since 1/12 of the rectangle is shaded,

Ratio of Rectangle area to shaded area = 12 : 1

Make the units the same for the rectangle by changing them to a common multiple.

The lowest common multiple of 9 and 12 is 36 (9 x 4 = 12 x 3 = 36)

So,

Area of triangle : area of rectangle
= 4 : 9
= 16 : 36 (multiply by 4)


Area of rectangle : shaded area
= 12 : 1
= 36 : 3 (multiply by 3)

Combined,

Area of triangle : area of rectangle : shaded area = 16 : 36 : 3

3 units = 36 cm²
1 unit = 36 cm² ÷ 3 = 12 cm²

Area of triangle = 16 units = 12cm² × 16
= 192cm²

Area of unshaded part

= area of triangle - 1 shaded area + area of rectangle - 1 shaded area

= area of triangle + area of rectangle - 2 shaded areas

= 16 units + 36 units - 2 × 3 units

= 52 units - 6 units

= 46 units

= 46 × 12 cm²

= 552 cm²