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 sum of their present ages.

Answer 1

Hint: We can solve this question by converting statements to equations and also we always let the present age.

Complete step-by-step solution -
Let the age of Nisha is x
Given: Asha's age is two more than her daughter’s age.
Hence Asha’s age is ${{x}^{2}}+2$
When Nisha grows up to her mother’s present age. So we can find that time by subtracting Nisha’s age at that time and present age.
So time till when Nisha grows up to her mother’s present age is
$\Rightarrow {{x}^{2}}+2-x$
Hence Asha’s age after ${{x}^{2}}+x-2$ years is
$\Rightarrow {{x}^{2}}+2+{{x}^{2}}+2-x$
$\Rightarrow 2{{x}^{2}}-x+4$
According to the question at this time
Asha’s age = 10 times of Nisha’s age – 1
$\Rightarrow 2{{x}^{2}}-x+4=10x-1$
$\Rightarrow 2{{x}^{2}}-x+4-10x+1=0$
$\Rightarrow 2{{x}^{2}}-11x+5=0$
$\Rightarrow 2{{x}^{2}}-10x-x+5=0$
$\Rightarrow 2x(x-5)-1(x-5)=0$
$\Rightarrow (x-5)(2x-1)=0$
Either $x=5$ or $x=\dfrac{1}{2}$
But age should be a whole number.
So Nisha’s present age is 5 years.
As we let Asha’s present age is ${{x}^{2}}+2$
On substituting x = 5
$\Rightarrow {{5}^{2}}+2=27$
Hence Asha’s present age is 27 years.

Note: We can do this question in terms of two variables. We can let Asha’s and Nisha’s ages as different variables. Then we will get two equations in terms of two variables which we can solve either by substitution method or elimination method. but finding the values of two variables will be lengthy.