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Try this beautiful problem from the Pre-RMO, 2017, Question 25, based on Area of a triangle.

Area of triangles – PRMO 2017, Question 25

Let ABCD be a rectangle and let E and F be points on CD and BC respectively such that area (ADE) =16, area (CEF) =9, and area (ABF) =25. What is the area of triangle AEF?

• is 107
• is 30
• is 840
• cannot be determined from the given information

Key Concepts

Equation

Algebra

Integers

But try the problem first…

Source

PRMO, 2017, Question 25

Higher Algebra by Hall and Knight

Try with Hints

First hint

Let x=AD, a=DE, y-a=EC, b=FC, x-b=FB, y=AB

$xa=32, b(y-a) =18, y(x-b) =50$

or, $by-\frac{32b}{x}=18$

or, $b=\frac{18b}{xy-32}$

Second Hint

or, $xy-\frac{18xy}{xy-32}=50$

or, $xy=t$

or, $\frac{t^{2}-32t-18t}{t-32}=50$

Final Step

or, $t^{2}-50t=50t-1600$

or, $t^{2}-100t+1600=0$

or, t=80,20

or, xy=80

or, area of triangle AEF= 80-(16+9+25)=30.