Question

# At present Asha’s age (in years) is 2 more than the square of her daughter Nisha’s age. When Nisha grows to her mother’s present age. Asha’s age would be one year less than 10 times the present age of Nisha. Find the present ages of both Asha and Nisha.

Hint:- Let us write equations for the given conditions and then find the age of Nisha and her mother by solving the equations.

Let the present age of Asha will be x years.
And the present age of Nisha will be y years.
So, according to the present age condition given for Nisha and her mother.
Square of Nisha present age will be y*y = ${{\text{y}}^{\text{2}}}$.
So, Asha age will be 2 more than the square of Nisha age.
So, x = ${{\text{y}}^{\text{2}}}$ + 2 (1)
And the difference in the ages of Nisha and her mother is (x – y) years.
As we know that difference in their present age is (x – y). So, after (x – y) years Nisha’s age will become equal to the present age of her mother.
So, after (x – y) years.
Nisha age will be y + (x – y) = x years.
And, Asha age will be x + (x – y) = 2x – y years.
And according to the question, when Nisha is the present age of her mother (x years) then Asha age would be one year less than 10 times the present age of Nisha.
So, Asha age will be (10y – 1) years.
But we had above found that Asha age would be (2x – y) years.
So, 2x – y = 10y – 1.
Now solving the above equation to get the value of x.
2x = 11y – 1
x = $\dfrac{{{\text{11y - 1}}}}{{\text{2}}}$ (2)
Now putting the value of x in equation 1. We get,
$\dfrac{{{\text{11y - 1}}}}{{\text{2}}}$ = ${{\text{y}}^{\text{2}}}$ + 2
On, cross-multiplying above equation.
11y – 1 = 2${{\text{y}}^{\text{2}}}$ + 4
2${{\text{y}}^{\text{2}}}$ – 11y + 5 = 0 (3)
Now, we had to solve equation 3 to get the value of y.
On, splitting middle term 11y. We get,
2${{\text{y}}^{\text{2}}}$ – 10y – y + 5 = 0
Now making the factor of the above equation to get the value of y.
2y (y – 5) –1(y – 5) = 0
Taking (y – 5) common from the above equation. We get,
(y – 5)(2y – 1) = 0
From above equation y = 5 years or y = $\dfrac{{\text{1}}}{{\text{2}}}$years or 6 months (impossible because in months).
Now putting the value of y in equation 2. We get,
x = $\dfrac{{{\text{11(5) - 1}}}}{{\text{2}}}$ = $\dfrac{{{\text{54}}}}{{\text{2}}}$ = 27 years
Hence, the present age of Nisha is 5 years and the present of Asha is 27 years.
Note:- Whenever we come up with this type of problem then first, we assume that their present ages are x and y years. And after that we had to find the equations of their ages according to the given conditions in question. After that we will solve the given equations to get appropriate values of x and y. And if any of the equations is quadratic then we have to neglect impossible values of x and y.