Question

# Sudesh is twice as old as Seema. If six years is subtracted from Seema’s age and four years are added to Sudesh’s age, Sudesh will be four times Seema’s age. How old were they three years ago?

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Hint: Take any two variables representing Sudesh’s age and Seema’s age. Form linear equations based on the data given in the question. Solve those equations by elimination method to calculate their exact ages.

We have to find Sudesh’s and Seema’s age based on the data given in the question. Let’s assume Sudesh’s age is $x$ years and Seema’s age is $y$ years.
We know that Sudesh is twice as old as Seema. So, we have $x=2y.....\left( 1 \right)$.
Six years ago, Seema’s age was $y-6$ years. Four years later, Sudesh’s age will be $x+4$ years.
As six years subtracted from Sudesh’s age is four times four years added to Seema’s age, we have $x+4=4\left( y-6 \right)$.
Simplifying the above equation, we have $x=4y-28.....\left( 2 \right)$.
We have two linear equations involving the variables $x$ and $y$.
Substituting equation $\left( 1 \right)$ in equation $\left( 2 \right)$, we have $2y=4y-28$.
Simplifying the above equation, we have $2y=28$.
$\Rightarrow y=14$
Substituting the value $y=14$ in equation $\left( 1 \right)$, we have $x=2y=2\left( 14 \right)=28$.
Thus, Sudesh’s age is $x=28$ years and Seema’s age is $y=14$ years.
We have to find their ages three years ago. Subtracting $3$ from their ages, we have Sudesh’s age $3$ years ago $=28-3=25$ years and Seema’s age $3$ years ago $=14-3=11$ years.
Hence, Sudesh’s age $3$ years ago is $25$ years and Seema’s age $3$ years ago is $11$ years.

Note: We can also solve this question by forming linear equations in one variable. We can take Sudesh’s age to be $x$ years and write Seema’s age in terms of Sudesh’s age based on given data and solve the equations to get the exact ages. One must keep in mind that we are required to find their ages three years ago. So, it’s necessary to subtract three from their present ages to get the correct answer.