Require ratio is 25:28. Explanation as follows.

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Draw lines *dm* and *ne* parallel to *bc*.

?*ane* and ?*abc* are similar. For similar triangles, the ratio of areas is square of the ratio of linear dimensions.

Since *ae*:*ec* is 3:5, *ae*:*ac* = 3:8.

So ratio of A(?*ane*) and A(?*abc*) = 9:64

Similarly, ratio of A(?*adm*) and A(?*abc*) = 4:49

Taking LCM of numbers, ratio of A(?*adm*), A(?*ane*) and A(?*abc*) = 256:441:3136

Now to find the area of quadrilateral *bdec* we need to calculate A(?*dne*).

Consider ?*adm* and ?*ane*. These are similar triangles. So ratio of *dm* and *ne* is same as the ratio of the square roots of the areas.

? Ratio of A(?*adm*) and A(?*ane*) = 256:441

⇒ Ratio of *dm* and *ne* = 16:21

Consider ?*dme* and ?*dne*. Their areas are divided in the ratio of the *dm*:*ne* = 16:21

The sum of the A(?*dme*) and A(?*dne*) is difference in the A(?*adm*) and A(?*ane*). Since A(?*adm*) and A(?*ane*) are in the ratio 256:441, the A(?*dmne*) is proportional to 185.

Out of this, A(?*dne*) = 21*185/(16+21) = 105

So A(?*ade*) is proportional to difference of A(?*ane*) and A(?*dne*).

So A(?*ade*) is proportional to 336.

Now A(?*bdec*) is difference of A(?*abc*) and A(?*ade*). So A(?*bdec*) is proportional to 2800.

So ratio of A(?*bdec*) to A((?*abc*) = 2800:3136 = 700:784 = 175:196 = 25:28

Hence, required ratio is 25:28.

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