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Sum of Coefficients of All Terms

If x3+3xy2/3x...

Question

If $\frac{x^{3} + 3 x y^{2}}{3 x^{2} y + y^{3}} = \frac{m^{3} + 3 m n^{2}}{3 m^{2} n + n^{3}}$ , show that : nx = my.
$[3 M a r k s]$

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Solution

$fraction numerator x cubed plus 3 x y squared over denominator 3 x squared y plus y cubed end fraction equals fraction numerator m cubed plus 3 m n squared over denominator 3 m squared n plus n cubed end fraction A p p l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o fraction numerator x cubed plus 3 x y squared plus 3 x squared y plus y cubed over denominator x cubed plus 3 x y squared minus 3 x squared y minus y cubed end fraction equals fraction numerator m cubed plus 3 m n squared plus 3 m squared n plus n cubed over denominator m cubed plus 3 m n squared minus 3 m squared n minus n cubed end fraction fraction numerator left parenthesis x plus y right parenthesis cubed over denominator left parenthesis x minus y right parenthesis cubed end fraction equals fraction numerator left parenthesis m plus n right parenthesis cubed over denominator left parenthesis m minus n right parenthesis cubed end fraction fraction numerator x plus y over denominator x minus y end fraction equals fraction numerator m plus n over denominator m minus n end fraction A p p l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o fraction numerator x plus y plus x minus y over denominator x plus y minus x plus y end fraction equals fraction numerator m plus n plus m minus n over denominator m plus n minus m plus n end fraction fraction numerator 2 x over denominator 2 y end fraction equals fraction numerator 2 m over denominator 2 n end fraction x over y equals m over n n x equals m y$ $[3 M a r k s]$

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