If the numerator of a fraction is increased by 2 and the denominator is increased by 1, the fraction because 5/8 and if the numerator of the same fraction is increased by 3 and the denominator is increased by 1, the fraction becomes 3/4 . What is the original fraction?

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

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

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

2/5 = 0.4 , 5/6 = 0.833, 11/12 = 0.916 and 7/8 = 0.875
Clearly, the greatest fraction is 0.916, i, e . 11/12.

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

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

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

Given expression = [(3²)² x (3 x 3 x 2)⁴] / 3¹⁶
= (3⁴ x 3⁸ x 2⁴) / 3¹⁶
= 2⁴ / 3⁴ = 16 / 81

Let man's salary be Rs. N
Money spend on food =N/7
Money spends on clothing = (N - N/7) x 1/2 = (6N/7) x (1/2) =3N/7
Money spend on entertainment = (N - N/7 - 3N/7)/3 = (7N - 4N)/7 = N/7
Now, money left =600
? N - N/7 - 3N/7 - N/7 = 600
? 7N - 5N/7 = 600
? N = 600 x 7/2 = Rs. 2100
? Money spend on entertainment = 1/7 x 2100 = Rs. 300

If the required fraction be N.
Then, (N x N) / (1/N) = 18²⁶/₂₇
? N³ = 512 / 27
? N = 8/3 = 2²/₃