Question

The difference between two positive integers is 36. The quotient when one integer is divided by another is 4. Find the integers.

Answer 1

Hint: We will find the two positive integers whose difference is 36 by supposing the first integer to be n and then the second number would be (n-36). It is said that when two integers are divided, we get 4 as the quotient. So, another relationship $\dfrac{n}{n-36}=4$ can be formed. We can find the two integers by solving these relations.

Complete step-by-step solution -
It is given in the questions that we have two positive integers whose difference is 36. So, we will start the solution by taking the first positive integer as n. Now, since the difference is given to us as 36, we can represent the second positive integer as (n-36). So, we have the first number as n and the second number as (n-36).
Now, we have to apply the second condition to the integers. It is given in the question that the quotient when one integer is divided by the other is 4. We can express the relation between the two positive integers as,
$\dfrac{n}{n-36}=4$
Cross multiplying the above equation, we get,
$\begin{align}
  & n=4\left( n-36 \right) \\
 & n=4n-144 \\
\end{align}$
Transposing $4n$ from the RHS to the LHS, we get,
$\begin{align}
  & n-4n=-144 \\
 & -3n=-144 \\
\end{align}$
On multiplying both sides with -1, we get,
$3n=144$
Now, on dividing both sides by 3, we get,
$\begin{align}
  & \dfrac{3n}{3}=\dfrac{144}{3} \\
 & n=48 \\
\end{align}$
Thus, we get our first positive integer as n = 48. So, now we can find the second positive integer given by (n-36). Therefore, we can substitute n = 48 and we will get, 48 – 36 = 12 as the second positive integer.
So, we have found out the two positive integers as 48 and 12.

Note: As it is given in the question that we have to consider only positive integers, we must be careful in taking the numbers. There is a possibility that we might assume the first integer as x and the second integer as x + 36 instead of x – 36 and this will lead us to the wrong answer. It is said in the question that the quotient is 4, so we can even take $\dfrac{n-36}{n}=4$ instead of $\dfrac{n}{n-36}=4$. Both ways are correct, we will get the integers as 12 and 48 instead of 48 and 12 in respective cases.