In the diagram below, we have PQ = 2AP = 2QB and AR/RC = 2/3. Find the ratio of the area of triangle BCQ to the area of triangle CQR.

Look at triangle(ABC)  --  Drop a perpendicular from C to AB.

This will be the height of triangle(CQB) with base QB.

It will also be the height of triangle(CQA) with base AQ.

Since AQ = 3 x QB, the area of triangle(CQA) = 3 x area of triangle(CQB).

 

Look at triangleAQC)  --  Drop a perpendicular from Q to AC.

This will be the height of triangle(QRA) with base RA.

It will also be the height of triangle(QRC) with base RC.

Since AR/RC = 2/3, the area of triangle(QRA) = 2/3 · area of triangle(QRC).

 

This means that triangle(QRC) = 3/5 the area of triangle(CQA).

 

Can you take it from here?