Question

At present suman's age is 7 more than a cube number and 12 less than its immediate next cube number.in how many years will suman be 40 years?

Answer 1

Let Suman's age = x and the number = y
So, x = 7 + y³
And the next no. to y = y + 1
So, x = 7 + y³ = (y + 1)³ - 12
So,
7 + y³ = (y + 1)³ -12
7 + y³ = y³ +1³ +3(y² × 1) + 3(y × 1²) -12 {Using identity: (a + b)³ = a³ + b³ + 3a²b + 3ab²}

7 + y³ = y³ + 1 + 3y² + 3y -12
7 +12 -1 = y³ - y³ + 3y² + 3y
18 = 3(y² + y)
6 = y² +y
y² + y - 6 = 0
y² + 3y - 2y - 6 = 0
y(y + 3) - 2(y + 3) = 0
(y - 2)(y + 3) = 0
Since, we want a positive number for prefering age,
y - 2 = 0
y = 2
So,
x = (2)³ +7 = 8 + 7 = 15
So, 40-15 = 25 more years will be needed for suman to become 40 years old.

Answer 2

Suman's age  S = 7 + N³      ,   let N be a positive integer
                       S = (N+1)³ - 12

  So   N³ + 7 = N³ + 3 N² + 3 N + 1  - 12
 
       =>  3 N² + 3 N - 18 = 0
     =>    N² + N - 6 = 0 ,                   -6 factors = 3 & -2  whose sum is +1.
     =>    (N - 2 ) (N + 3) = 0
                 N = 2  or - 3  , we ignore the negative value.
  
    Suman's age = 7 + 2³ = 15 yrs
   Sumans' age will be 40 years after 25 years