Question

Anushka and Aarushi are friends. They have an equal amount of money in their pockets. Anushka gave ${\left( {\dfrac{1}{3}} \right)^{rd}}$ of her money to Aarushi as her birthday gift. Then Aarushi gave a party at a restaurant and cleared the bill by paying half of the total money with her. If the remaining money in Aarushi's pocket is Rs.1600, find the sum gifted by Anushka.

Answer 1

Hint: We will first assume Anushka and Arushi has Rs. X in their pocket. After that we will apply the condition of Anushka giving ${\left( {\dfrac{1}{3}} \right)^{rd}}$ of her money to Aarushi. Now, we will calculate Arushi’s money and then equate its half to Rs. 1600.

Complete step by step answer:
Let us assume that Anushka and Arushi both have $Rs.x$ in their pocket in the start.
Now, it is Arushi’s birthday. So, Anushka gifted ${\left( {\dfrac{1}{3}} \right)^{rd}}$ of her money to Arushi.
This implies that Anushka gifts her $Rs.\dfrac{1}{3}x$. ………(1)
So, now Anushka is left with $Rs.\left( {x - \dfrac{1}{3}x} \right) = Rs.\dfrac{2}{3}x$.
And, Arushi will have $Rs.\left( {x + \dfrac{1}{3}x} \right) = Rs.\dfrac{4}{3}x$.
Now, since Arushi gave a party at a restaurant by half of the amount she, has now.
This implies that she spent $Rs.\dfrac{1}{2}\left( {\dfrac{4}{3}x} \right)$ that is $Rs.\dfrac{4}{6}x$ on the party.
So, she would be left with $Rs.\dfrac{4}{6}x$ with her.
And, we are given that she is left with Rs. 1600.
So, equating these data, we will see that $Rs.\dfrac{4}{6}x = Rs.1600$
Cross multiply this expression to get:-
$ \Rightarrow 4x = 6 \times 1600$
Simplifying the RHS, we will get:-
$ \Rightarrow 4x = 9600$
Taking the 4 from multiplication in LHS to division in RHS to get the following expression:-
$ \Rightarrow x = \dfrac{{9600}}{4}$
Simplifying the RHS to get:-
$ \Rightarrow x = 2400$
Therefore, they had Rs. 2400 each in their pockets.
Putting this in (1), we will get:-
Anushka gifts Arushi the amount $Rs.\dfrac{1}{3}x = Rs.\dfrac{{2400}}{3} = Rs.800$.

Hence, the required answer is Rs. 800.

Note:
The students must note that they must go in sequence using the data in the question. It will become really easy because this is just an equation in one variable.
The students must know that they need as many equations as many variable they have. Like in this question, we just had one variable, and we had one data of Rs. 1600 to solve the same.
Additional Information:- If we have equation in one variable, we need only the number line to solve it. If we have equation in two variable, we require coordinate axis with x-axis and y-axis to solve it.