Question

Seven years ago Varun’s age was five times the square of Swati’s age. Three years hence Swati’s age will be two fifth of Varun’s age. Find their present ages.

Answer 1

Hint: Take Varun and Swati’s present age as V and S respectively. Make equations according to the given condition. You will have two equations, now solve them simultaneously and get the result.

Let us assume that the present age of Varun and Swati be V and S respectively.

In the question we have been given conditions on their ages 7 years ago.

So, seven years ago the age of Varun and Swati were (V-7) and (S-7), respectively.

As per the given criteria, the relation between their ages will be,

$V-7=5{{\left( S-7 \right)}^{2}}$$$$$

Now using the formula ${{\left( x-y \right)}^{2}}={{x}^{2}}+{{y}^{2}}-2xy$, the above equation can be written as,

So, we can write it as,

\[V-7=5\left( {{S}^{2}}-14S+49 \right)\]

On opening the brackets, the above equation can be written as,

$V=5{{S}^{2}}-70S+252...........\left( i \right)$

So, we got a relation between V and S by first condition.

Now as another condition of three years hence is given then the present age of Varun and Swati will become (V+3) and (S+3) respectively.

So the relation is,

$\left( S+3 \right)=\dfrac{2}{5}\left( V+3 \right)$

On cross multiplication we will get,

5(S+3) = 2(V+3)

Now on further simplifying we get,

5S + 15 = 2V + 6

Now subtracting 6 to both sides of the above equation, we get

2V = 5S + 9…………….(ii)

Now we will substitute the relation of equation (ii) in equation (i) we get,

$2\left( 5{{S}^{2}}-70S+252 \right)=5S+9$

Now on further simplification we get,

$10{{S}^{2}}-145S+495=0$

Now we will do middle term factorization to factorize $10{{S}^{2}}-145S+495$ we will get,

$10{{S}^{2}}-90S-55S+495=0$

Further factoring we get,

10S(S-9) – 55(S-9) = 0

So it can be written as,

(10S-55) (S-9) = 0

10S=55 or S=9

So, the value of S can be $\dfrac{55}{10}$ or 9.

Only ‘9’ is possible as there was a condition given that we were comparing ages of about

 seven years ago hence the age should be greater than ‘7’.

So, the present age of Swati is 9 years.

Now substituting Swati’s age as 9 years in equation (ii) we get,

2V = (5(9)+9)

On simplifying we get,

2V =54

Hence the value of V is 27.

So, the present age of Varun is 27 years old.

 Hence, the present age of Varun and Swati is 27 and 9 years, respectively.


Note: Students should be careful while making equations and finding relations between Varun and Swati’s age. They should also take care about the calculations to avoid any mistakes.

Another approach is using the condition that seven years ago Varun’s age was five times the square of Swati’s age, considering Swati’s age and Varun’s age seven years ago as, ‘x’ and $'5{{x}^{2}}'$. Now use the second condition and find out the value of ‘x’. Be careful this is the age seven years ago. You have to add seven to it to get the present age. So this process is tedious and confusing.