If (7m + 2n)/(7m - 2n) = 5/3 ; use the properties of proportion to find:(i) m:n. (ⅱ) (m ^ 2 + n ^ 2)/(m ^ 2 - n ^ 2). [icse 2017]

(i) m:n 

Given that ,

(7m + 2n)/(7m - 2n) = 5/3

Applying componendo and dividendo, we get, ((7m + 2n) + (7m - 2n))/((7m + 2n) - (7m - 2n)) = (5 + 3)/(5 - 3)

[By using (Num. + Deno.)/(Num. - Deno.) on both sided]

=) (7m + 2n + 7m - 2n)/(7m + 2n - 7m + 2n) = 8/2 

=> 14m/4n = 8/2 => (7m)/(2n) = 4/1

Multiplying both sides by 2/7 , we get,

 2/7 * 7/2 * m/n = 2/7 * 4/1 

=> m/n = 8/7 => m:n = 8:7

(ii) Given that (7m + 2n)/(7m - 2n) = 5/3

Applying componendo and dividendo, we get, 

 ((7m + 2n) + (7m - 2n))/((7m + 2n) - (7m - 2n)) = (5 + 3)/(5 - 3)

=> (7m + 2n + 7m - 2n)/(7m + 2n - 7m + 2n) = 8/2

=> 14m/4n = 4 => 7/2(m/n) = 4/1 Multiplyning both sides by 2/7 , 

we get. m/n = 8/7

Squaring sides, we get (m²)/(n²) = 64/49

Applying componendo and devedando, we get,

m²+n²/m²-n² = 64+49/ 64-49

=> (m²+n²)/(m²-n²) = 113/15 Ans.