Question

The sum of four consecutive odd integers is $216$. What are the four integers?

Answer 1

Hint: We will consider the four consecutive but odd integers as \[x, x + 2,x + 4,x + 6\] . then on adding them we will equate it to 216 as it is given their sum in the question itself. We will get an equation with one variable. On solving it we will get the value of x. from that we can find the remaining three numbers.

Complete step by step answer:
Let the first odd integer be $x$. Now since the numbers are odd we can take or consider the remaining numbers as \[x + 2,x + 4,x + 6\]. Now it is given that their sum is 216. So we can write this as,
\[x + x + 2 + x + 4 + x + 6 = 216\]
On adding the variables separately we get,
\[4x + 12 = 216\]
Now taking the constant numbers one side,
\[4x = 216 - 12\]
On subtracting we get,
\[4x = 204\]
Transpose 4 on other side we get,
\[x = \dfrac{{204}}{4}\]
On dividing by 4 we get,
\[x = 51\]
Thus the first number we considered is 51. Now putting the value of x in the considered numbers we can find the remaining three numbers,
\[x + 2 = 51 + 2 = 53 \\
\Rightarrow x + 4 = 51 + 4 = 55 \\
\therefore x + 6 = 51 + 6 = 57 \\ \]
Thus the four consecutive odd numbers are \[51,53,55\,\text{and}\,57\].

Note: The numbers are consecutive and are odd. So we cannot consider the numbers as \[x,x + 1,x + 2,x + 3\] because these are consecutive but are not odd. So many students do get confused here only. But also note that if it is given consecutive even then also we can consider the numbers as \[x,x + 2,x + 4,x + 6\] because there is a difference of two in this series also.