Question

Ram, Shyam, Tarun and Varun together had the total amount of Rs.240 with them. Ram had half of the total amount with the other. Shyam had one-third of the total amount with others. Varun had one third of the total amount with others. Find the amount with Varun.

Answer 1

Hint: First, we have to consider variables like say suppose x, y, z, w for 4 people Ram, Shyam, Tarun and Varun which have a total amount of Rs. 240. Then, we have to find their individual amount by making an equation and then substituting values of variables found into equations for finding the answer. Let’s take as it is told that Ram, Shyam, Tarun and Varun together had the total amount of Rs.240 with them then in mathematical form it is written as: \[\text{ }x+y+z+w=240\] . Similarly, we have to make other equations also.

Complete step-by-step answer:
Here, we will consider that Ram has x rupees, Shyam has y rupees, Tarun has z rupees, and Varun has w rupees.
It is given that the total amount is 240 rupees. So, writing it in mathematical form, we get as
\[\Rightarrow x+y+z+w=240\] ………………………………..(1)
Next, it is given as Ram had half of the total amount with the others. Shyam, had one third of the total amount with the others. Tarun had one fourth of the total amount with the others.
So, writing mathematical form for above sentence, we get
\[\Rightarrow \text{ }x=\dfrac{\left( y+z+w \right)}{2}\] ………………………….(2)
\[\Rightarrow y=\dfrac{\left( x+z+w \right)}{3}\] ………………………(3)
\[\Rightarrow ~z=\dfrac{\left( x+y+w \right)}{3}\] ………………………….(4)
From equation (1), we will subtract x from both the sides so, we will get:
\[~\Rightarrow y+z+w=240-x\] ………………………….(5)
Again, subtracting y on both sides of equation (1), we get:
\[\Rightarrow x+z+w=240-y\] ………………………………(6)
Again, subtracting z on both sides of equation (1), we get:
\[\Rightarrow x+y+w=240-z\] ……………………………..(7)
Now, we will substitute the value of equation (5) in equation (2), we will get
\[\Rightarrow \text{ }x=\dfrac{\left( 240-x \right)}{2}\]
\[\Rightarrow ~2x+x=240\]
\[\Rightarrow ~3x=240\]
\[\Rightarrow ~x=80\]
Therefore, Ram had 80 rupees.
Now, we will substitute the value of equation (6) in equation (3), we will get
\[\Rightarrow y=\dfrac{\left( 240-y \right)}{3}\]
\[\Rightarrow ~3y+y=240\]
\[\Rightarrow ~4y=240\]
\[\Rightarrow ~~y=60\]
Therefore, Shyam had 60 rupees.
Now, we will substitute the value of equation (7) in equation (4), we will get
\[\Rightarrow ~z=\dfrac{\left( 240-z \right)}{3}\]
\[\Rightarrow ~3z+z=240\]
\[\Rightarrow ~4z=240\]
\[\Rightarrow ~z=60\]
Therefore, Tharun had 60 rupees.
Now, we will substitute the values obtained of x, y, z in equation (1) we get,
\[\Rightarrow ~80+60+60+w=240\]
\[\Rightarrow 200+w=240\]
\[\Rightarrow w=240-200\]
\[\Rightarrow w=40\]
Therefore, the total amount with Varun is 40 rupees.
Hence, we get amounts Ram, Shyam, Tarun and Varun have as 80 Rs., 60 Rs., 60 Rs., 40 Rs. Respectively.

Note: Be careful while making the equation. Students make mistakes in understanding the equation given in sentence form. For example: Ram had half of the total amount with the other, this sentence is given. So, assuming Ram has x rupees. So, don’t write the equation as \[\dfrac{1}{2}x+y+z+w=240\] instead of writing \[x=\dfrac{\left( y+z+w \right)}{2}\] . Whole problem will lead to the wrong answer. So, be careful with it.