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If the sum of two consecutive odd numbers is \[312\] , find the numbers.
A) \[155\] and \[157\]
B) \[155\] and \[156\]
C) \[156\] and \[157\]
D) \[154\] and \[153\]

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Last updated date: 03rd Mar 2024
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IVSAT 2024
Answer
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Hint: Odd numbers are those numbers which are not completely divided by \[2\]. Here, we will solve this question by making an equation with a variable \[x\] .

Complete step-by-step solution:
Step 1: Let's assume that \[x\] is the first odd number and the next consecutive odd number will be equal to \[x + 2\].
For example, if the first odd number is \[1\] then the next consecutive odd number will be equal to \[3\].
Step 2: As given in the question that the sum of two consecutive odd numbers is \[312\], so we can write the equation as below:
\[x + \left( {x + 2} \right) = 312\]
By opening the brackets into the LHS side, we get:
\[ \Rightarrow x + x + 2 = 312\] ………. (1)
By adding the coefficient \[x\] into the LHS side of equation (1), we get:
\[ \Rightarrow 2x + 2 = 312\]
By bringing \[2\] into the RHS side of the equation \[2x + 2 = 312\] , and subtracting it from \[312\]:
\[ \Rightarrow 2x = 312 - 2\]
\[ \Rightarrow 2x = 310\]
By bringing \[2\] into the RHS side of the equation \[2x = 310\] and dividing it by \[310\] , we get:
\[ \Rightarrow x = \dfrac{{310}}{2}\]
\[ \Rightarrow x = 155\]
So, the second consecutive odd number will be equals to \[x + 2\], by substituting the value of \[x\] in it, we get:
\[ \Rightarrow 155 + 2 = 157\]

\[\therefore \] The two consecutive odd numbers are \[155\] and \[157\]. Option A is correct.

Note: In these types of questions, students need to take care while taking the two-consecutive odd/even numbers in the form of variables. It may lead to the wrong answer.
Also, we can apply short methods if the options are given in these questions by simply adding or subtracting, or multiplying them whichever is asked. But sometimes this technique can also fail so better go with the equation making method.
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