Question

Arun and Prabhat have some books with them. Once Prabhat said to Arun that, if Arun gives 3 books to Prabhat then Arun will have only \[\dfrac{1}{2}\] of the books that Prabhat will have with him. Then Arun asked frankly that if Prabhat gives him only two books (to Arun), then Prabhat will have as many books as Arun will have. The total number of books that Arun and Prabhat have with them is _______.
(a) 25
(b) 56
(c) 30
(d) Can’t be determined

Answer 1

Hint: In this question, take two separate variables to determine the number of books Arun and Prabhat have. Solve to find x and y and the summation of x and y will be the total number of books they have.

Complete step-by-step solution -
Let Arun have x number of books and let Prabhat have y number of books. We can observe that there are two conditions given in the question. Therefore, the first condition is:
\[\left( x-3 \right)=\left( y+3 \right)\dfrac{1}{2}\]
On solving, we get,
\[\Rightarrow 2x-6=y+3\]
\[\Rightarrow 2x-y=3+6\]
\[\Rightarrow 2x-y=9.....\left( i \right)\]
The second condition is,
\[x+2=y-2\]
\[\Rightarrow x-y=-4\]
\[\Rightarrow y=x+4....\left( ii \right)\]
On substituting (ii) in (i), we get,
\[2x-\left( x+4 \right)=9\]
\[\Rightarrow 2x-x-4=9\]
\[\Rightarrow x=9+4=13\]
\[\Rightarrow x=13....\left( iii \right)\].
So, Arun has 13 books.
For finding the value of y we put the value of x in equation (ii)
So the value of y is y=$x+4$ = $13+4$ =17
As We have assumed that Prabhat has y number of books. So, Prabhat has 17 books
Hence the total number of books that Arun and Prabhat will have $x+y$ = $13+17$ =30.

Note: An alternate way of solving this question with two conditions is to write both equations in the form of ‘ax + by = c’ and then solve them simultaneously by adding, or subtracting and get the value of a variable and then substitute this value in any one of the simplified forms of the condition to get the value of the other variable and then adding these two values to get the required answer. In these types of questions, always remember to read the question carefully and form proper conditions. So, if the conditions are wrong, the whole solution will be incorrect.