Question

Find three consecutive odd numbers whose sum is 219.

Answer 1

Hint:
Here we need to find the value of three consecutive odd numbers. For that, we will assume these three consecutive numbers to be any variable. Then we will add these three assumed consecutive odd numbers to get the sum. Then we will equate the obtained sum with the given sum to get the value of the variables and hence the value of each number.

Complete step by step solution:
Let’s assume the three consecutive odd numbers be \[a\] , \[a + 2\], \[a + 4\]
Now, we will find the sum of these three consecutive odd numbers by adding them.
Sum of three consecutive numbers \[ = a + a + 2 + a + 4\]
On adding the like terms, we get
\[ \Rightarrow \] Sum of three consecutive numbers \[ = 3a + 6\] …….. \[\left( 1 \right)\]
It is given that the sum of these three consecutive numbers is equal to 219.
Equating the obtained sum with the given sum, we get
\[ \Rightarrow 219 = 3a + 6\]
On subtracting 6 from both sides, we get
\[\begin{array}{l} \Rightarrow 219 - 6 = 3a + 6 - 6\\ \Rightarrow 213 = 3a\end{array}\]
Dividing both the sides by 3, we get
\[\begin{array}{l} \Rightarrow \dfrac{{213}}{3} = \dfrac{{3a}}{3}\\ \Rightarrow a = 71\end{array}\]
Now we will substitute the value of \[a\] in \[a + 2\] and \[a + 4\], to get the second and third number.
So the second number, \[a + 2 = 71 + 2 = 73\]
The third odd number, \[a + 4 = 71 + 4 = 75\]

Hence, the three consecutive odd numbers are 71, 73 and 75.

Note:
Here, we assumed the variable to be \[a\] , \[a + 2\], \[a + 4\], and not \[a\] , \[a + 1\], \[a + 2\] because we had to find consecutive odd numbers and not just three consecutive numbers. In two consecutive odd numbers there is a difference of 2 but if we had chosen \[a\] , \[a + 1\], \[a + 2\] we get the difference as 1 which is one. If we had to find the consecutive even number instead of consecutive odd numbers then we would have chosen the variables \[a\] , \[a + 1\], \[a + 3\].