Question

Find two consecutive odd positive integers whose sum is 80.

Answer 1

Hint: Here we assume two odd positive integers and then form up an equation of sum of these two numbers and equate it to 80. First few odd numbers are of the form \[2n + 1,2n + 3,2n + 5....\]

Complete step-by-step answer:
We know that odd integers are of the form \[2n + 1,2n + 3,2n + 5....\]
The difference between two integers is
 \[(2n + 3) - (2n + 1) = 2n + 3 - 2n - 1 = 2\]
So if we assume the first odd positive integer as \[x\]
Then consecutive odd positive integer will be \[x + 2\]
Now we know the sum of two consecutive odd integers is 80.
\[
   \Rightarrow x + x + 2 = 80 \\
   \Rightarrow 2x + 2 = 80 \\
 \]
Shift all the constant values to one side
\[
   \Rightarrow 2x = 80 - 2 \\
   \Rightarrow 2x = 78 \\
 \]
Dividing both sides by 2
\[
   \Rightarrow \dfrac{{2x}}{2} = \dfrac{{78}}{2} \\
   \Rightarrow x = 39 \\
 \]
For second consecutive integer \[ \Rightarrow x + 2 = 39 + 2 = 41\]
So the first number is 39 and the second number is 41.

Note: Students many times make the mistake of not changing the sign when shifting the values from one side of the equation to another side, always keep in mind the sign changes from positive to negative and vice versa when we shift a number from one side to another side of the equation.
Students also make the mistake of considering \[(2n - 1)\] as the odd number which is wrong because if we put the value of as 0 we will get a negative odd integer and in this question we are taking values of positive odd integers.
Alternate method:
We know that odd integers are of the form \[2n + 1,2n + 3,2n + 5....\]
Considering two consecutive odd integers \[2n + 1,2n + 3\]
Then forming equation of sum of two numbers
\[
   \Rightarrow 2n + 1 + 2n + 3 = 80 \\
   \Rightarrow 4n + 4 = 80 \\
 \]
Shift all constants to one side of the equation
\[
   \Rightarrow 4n = 80 - 4 \\
   \Rightarrow 4n = 76 \\
 \]
Divide both sides of the equation by 4
\[
   \Rightarrow \dfrac{{4n}}{4} = \dfrac{{76}}{4} \\
   \Rightarrow n = 18 \\
 \]
So substituting the value of n to the two numbers we get
\[
   \Rightarrow 2n + 1 = 2(19) + 1 = 38 + 1 = 39 \\
   \Rightarrow 2n + 3 = 2(19) + 3 = 38 + 3 = 41 \\
 \]
So two consecutive odd positive integers are 39 and 41.