The greatest fraction out of 2/5, 5/6, 11/12 and 7/8 is?

Given Expression = (6.5 x 4.7 + 6.5 x 5.3 ) / (1.3 x 7.9 - 1.3 x 6.9)
= [6.5 x (4.7 + 5.3 )] / [13 x (7.9 - 6.9)]
= (6.5 x 10) / (1.3 x 1)
= 50

Given expression = [(3.537 - .948)² + (3.537 + .948)²] / [(3.537)² + (.948)²]
= [(a - b)² + (a + b)²] / (a² + b²)

(Where a = 3.537 and b = 0.948)
= 2(a² +b²) / (a² +b²)
= 2

Given expression = (.23 - .023) / ( .0023 / 23 )
= .207 / (.0023 / 23)
= .207 / .0001
=.2070 / .0001
= 2070

Given expression = ?0.5 x .5 x a = .5 x .05 x ? b
? ?.025 x a = 0.25 x ? b
? 0.025a = 0.25 x .025 x b
? a/b = (.025 x .025) /.025
= .025

a / b = 0.04 / 1.5
= 4 / 150
= 2 / 75

? Given Exp. (b - a) / ( b + a)
= (1- a/b ) / (1 + a/b)
= (1- 2 / 75) / (1 + 2 / 75)
= ( 73/75 ) / ( 77/75)
= 73/77

Given expression =(.6)³ + (.4) ³ + 3 x .6 x .4 x (.6 + .4)
= a³ + b³ + 3ab (a+b)
= (a+b)³
= (.6 + .4)³
= 1³
= 1

Given Expression = (.53)³ - (.42)³- 3 x .58 x .42 x (.58-.42)
= a³-b³ -3ab(a - b)
= (a - b)³

(where a = .58 and b=.42)
= (.58 - .42 )³
= .16 x .16 x .16
= 0.004096

Let the required fraction be p/q
? p / (q + 3) = 1/3
? 3p - q = 3 ...(i)

and, (p + 4) / q = 3/4
? 4p - 3q = -16 ...(ii)

Solving these equations, we get p = 5, q = 12
? Required fraction = 5/12

Let the original fraction be p/q
Then, (p + 2) / (q + 1) = 5/8
? 8p - 5q = 11 .....(i)
Again (p + 3) / (q + 1) = 3/4
? 4p - 3q = 9 ...(ii)
Solving equations , (i) and (ii), we get ,
p = 3 and q = 7
? Fraction = 3/7

Let the numerator and denominator be p and q respectively.
Then, (p + 2) / (q + 3) = 7/9
? 9(p + 2) = 7(q +3)
? 9p - 7q = 3 ....(i)
Again, (p - 1) / (q - 1) = 4/5
? 5p - 4q = 1 ....(ii)

Solving (i) and (ii) we get,
p = 5, q = 6
Required fraction = 5/6