Question

Aishwarya is twice as old as Karishma. If six years are subtracted from Karishma’s age and four years are added to Aishwarya’s age, then Aishwarya’s age will be $4$ times Karishma’s age. How old were they two years ago?

Answer 1

Hint: Assume the current age of Aishwarya and Karishma to be x and y respectively . Then, form two equations which show relation between age of Aishwarya and karishma according to the first two statements. Solve both equations and subtract two years from the values obtained.

Complete step-by-step answer:
Let us assume the age of Aishwarya is y and the age of karishma is x.
Now given that Aishwarya is twice as old as Karishma then we can write,
$ \Rightarrow y = 2x$ -- (i)
Now if six years are subtracted from karishma’s age then her age is=$x - 6$
And four years are added to aishwarya’s age the her age is=$y + 4$
Then according to question, Aishwarya’s age will be $4$ times Karishma’s age
$ \Rightarrow y + 4 = 4\left( {x - 6} \right)$
On opening the bracket and multiplying, we get
$ \Rightarrow y + 4 = 4x - 4 \times 6$
On solving we get,
$ \Rightarrow y + 4 = 4x - 24$
On taking variables one side and constants the other side we get,
$ \Rightarrow y - 4x = - 24 - 4$
On solving we get,
$y - 4x = - 28$ -- (ii)
Now on substituting the value of y from eq. (i) to eq. (ii), we get-
$ \Rightarrow 2x - 4x = - 28$
On subtraction we get,
$ \Rightarrow - 2x = - 28$
On cancelling the negative sign from both side we get,
$ \Rightarrow 2x = 28$
On transferring $2$ from left to right side, we get-
$ \Rightarrow x = \dfrac{{28}}{2}$
On division we get,
$ \Rightarrow x = 14$
On putting this value in eq. (i) we get,
$ \Rightarrow y = 2 \times 14 = 28$
So the current age of Aishwarya is $28$ and current age of Karishma is $14$
Now we have to find their ages two years ago so we will subtract $2$ from their current ages.
So the age of Aishwarya two years ago is=$28 - 2 = 26$
The age of karishma two years ago is = $14 - 2 = 12$
The age of Aishwarya and Karishma two years ago were $26$ and $12$ respectively.

Note: Here we used a substitution method to solve the two equations where we used value from one equation and substituted in the second equation to find the value of both variables. You can also use elimination methods by writing first eq. as-$y - 2x = 0$ and then eliminating y from both equations by subtracting eq. (i) from (ii).Then simplify the equation to get the value of x.