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 age of both Asha and Nisha.​

Answer 1

Let Nisha's present age be x years.
Then, Asha's present age $=(x^2+2)$ years.
when Nisha grows to her mother's present age i.e., after passing $\left\{\right(x^2+2)-x\}$ yr.
Then, Asha's age would be increased by $\left\{\right(x^2+2)-x\}$ yr,
i.e., $(x^2+2)-x+x^2+2$
$=\left\{2\right(x^2+2)-x\}$
Again by given condition,
Age of Asha = one year less than 10 the present age of Nisha,
$2(x^2+2)-x=10x-1$
$\Rightarrow 2x^2+4-x=10x-1$
$\Rightarrow 2x^2-11x+5=0$
On splitting the middle term, we get:
$2x^2-(10x+x)+5=0$
$\Rightarrow 2x^2-10x-x+5=0$
$\Rightarrow 2x(x-5)-1(x-5)=0$
$\Rightarrow (x-5)(2x-1)=0$
$\therefore x=5\text{or}x=\frac{1}{2}$
But $x\ne \frac{1}{2},$ as age should be a whole number.
Hence, required age of Nisha = 5 years
and required age of Asha $=x^2+2=(5^2+2)$
(=25+2=27) yrs.
Thus, present ages of Nisha and her mother Asha are 5 and 27 years respectively.