Question

In X class, if three students sit on each bench, one student will be left. If four students sit on each bench, one bench will be left. Find the number of students and the number of benches in that class.

Answer 1

Hint: Let the number of benches and students be x and y, respectively. It is given that if three students sit on each bench, one student will be left, so we can deduce that the number of students is one more than three times the number of benches, and this can be mathematically represented as y-3x=1 . Similarly use the other statement given to get another relation between x and y. Finally solve the equations to get the answer to the above question.

Complete step by step solution:
Let us start the solution to the above question by letting the number of benches and students be x and y, respectively.
Now, it is given that if three students sit on each bench, one student will be left, so we can deduce that the number of students is one more than three times the number of benches, i.e. y is one more than 3x. So, if we represent this in form of equation, we get
$y-3x=1......(i)$
Also, it is given that if four students sit on each bench, one bench will be left, which means that the number of students is 4 less than four times the number of students. This can be mathematically represented in terms of x and y as:
 $4x-y=4......(ii)$
Now we will add equation (i) and equation (ii). On doing so, we get
$y-3x+4x-y=1+4$
Cancelling y in the LHS and adding the other terms, we get
x=5
Now we will substitute the value of x in equation (i). On doing so, we get
$\begin{align}
  & y-15=1 \\
 & \Rightarrow y=16 \\
\end{align}$
Hence, the number of benches is 5 and the number of students is 16.

Note: The main thing to keep in mind is that while interpreting the second statement, one bench doesn’t mean that 4 times the number of benches is one more than the number of students, but as 4 are sitting one bench, it must be interpreted as 4 times the number of benches is four more than the number of students. The other thing to keep in mind is that you don’t get confused between the variables assigned to benches and students and form the wrong equation, by assuming x to be benches in first and y to be benches while forming the second equation.