Question:A fraction becomes `2/3` when 1 is added to both, its numerator and denominator. And, it become `1/2` when 1 is subtracted from both the numerator and denominator. 

Let the required fraction be `x/y` then,

 `(x + 1)/(y + 1)`=`2/3`

 => 3x - 2y = -1  ...........(i)

 And, `(x -1)/(y - 1)`= `1/2`

 => 2x -y = 1  ...........(ii)

 Solving (i) & (ii), we get x = 3, y = 5.

 `:.`Required fraction =`3/5`

Question:The ratio of the ages of Mona and Sona is 4:5. Twelve years hence, their will be in the ratio of 5:6. What will be Sona's age after 6 years? 

Let their present ages ages be 4x & 6x

Then `(4x + 12)/(5x + 12)`=`5/6`

or x = 12

`:.` Sona's age after 6 years = (5x + 6) = 66 years

Question:Ramu was 4 times as old as his as his son 8 years age. After 8 years, Ramu will be twice as old his son. What are their present ages? 

Let son's age 8 years ago be x years.

 Then, Ramu's age at that time = 4x years.

 Son's age after 8 years = (x + 8) + 8 = (x + 16) years.

 Ramu's age after 8 years = (4x + 8) + 8 =(4x + 16) years.

 `:.` 2 (x + 16) = 4x + 16 

 or x = 8

 `:.` Son's present age = (x + 8) = (8 + 8) =16 years.

 `:.` Ramu's present age =(4x + 8) =(4 x 8 +8) =40 years.

Question:A man is four times as old as his son. Five years ago, the man was nine times as old as his son was at that time. What is the present age of the man? 

Let son's age = x.

 Then, man's age = 4x.

 9 (x -5) = (4x - 5)

 or 9x - 45 = 4x - 5

 or 5x = 40

 or x = 8

`:.` Man's present age = 4x = 4 x 8 = 32 years.

Question:The sum of the ages of Aruna and her mother is 49 years. Also, 7 years ago, the mother's age was 4 times Aruna's age. Find the present age of Aruna's mother. 

Let Aruna's age 7 years ago be x.

 Mother's age 7 years ago = 4x

`:.` (x + 7) + (4x + 7) = 49

 or x =7

`:.` Mother's present age = (4x +7) = (4 x 7 +7) =35 years

Question:The ages of A and B differ by 16 years. If 6 years ago, the elder one be 3 times as old as the younger one, find their present ages. 

Let A's age = x & B's age = (x + 16)
3 (x - 6) = (x + 16 -6) 
or x = 7

`:.` Mother's present age = (4x + 7) = 35 years.

Question:`81^(3/4)` 

`(3^4)^(3/4)` = `3^(4 xx 3/4)` =`3^3` = 27

Question:`(1/64)^(-5/6)` 

`(1/64)^(-5/6)`

=`64^(5/6)`

=`(2^6)^(5/6)`

=`2^(6 xx 5/6)`

=`2^5`

=32

Question:`256^(-1/4` 

`256^(-1/4)`

=`(1/256)^(1/4)`

=`[(1/4)^4]^(1/4)`

=`(1/4)^(4 xx 1/4)`

=`1/4`

Question:`(x^b/x^c)^ ((b^2 + c^2 + bc))` x `(x^c/x^a)^ ((c^2 + a^2 + ca))` x `(x^a/x^b)^ ((a^2 + b^2 + ab))` 

=`(x^(b - c))^((b^2 + c^2 + bc))` x `(x^(c - a))^((c^2 + a^2 + ca))` x `(x^(a - b))^((a^2 + b^2 + ab))`

=`x^((b - c)(b^2 + c^2 + bc))`    x  `x^((c - a)(c^2 + a^2 + ca))`    x  `x^((a - b)(a^2 + b^2 + ab))`

= `x^((b^3 - c^3))`  x `x^((c^3 - a^3))`  x `x^((a^3 - b^3))`

=`x^((b^3 - c^3) + (c^3 - a^3) + (a^3 - b^3))`

=`x^0`

=1