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Algebra Level 3

If x : y : z = 2 : 3 : 4 x:y:z=2:3:4 , find the ratio x 3 + y 3 + z 3 − 3 x y z ( x + y + z ) 3 \dfrac{x^3+y^3+z^3-3xyz}{(x+y+z)^3} .

If the ratio is in the form p q \frac { p }{ q } , where gcd ⁡ ( p , q ) = 1 \gcd(p,q)=1 , find p + q p+q


The answer is 28.

Swapnil Das by Swapnil Das , 20, India

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1 solution

Since x : y : z = 2 : 3 : 4 x:y:z=2:3:4 , we get x = 2 t , y = 3 t , z = 4 t x=2t, y=3t, z=4t .

So, x 3 + y 3 + z 3 − 3 x y z ( x + y + z ) 3 = 8 t 3 + 27 t 3 + 64 t 3 − 72 t 3 729 t 3 = 1 27 \dfrac{x^3+y^3+z^3-3xyz}{(x+y+z)^3}=\dfrac{8t^3+27t^3+64t^3-72t^3}{729t^3}=\dfrac{1}{27} .

Thus, p + q = 1 + 27 = 28 p+q=1+27=\boxed{28} .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Please check your solution it should be 8 t 3 \boxed{8t^3} in your second last step.

Atanu Ghosh - 5 years, 2 months ago

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Correct it brother!

Atanu Ghosh - 5 years, 2 months ago

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