Question

If \[2n\] is an even number. What are the odd numbers on each side of it? The sum of two consecutive odd numbers is \[96\] What are they?
(a) \[41,43\]
(b) \[57,59\]
(c) \[47,49\]
(d) \[31,33\]

Answer 1

Hint: In order to solve this question, we will first try to see what are the odd numbers on each side \[2n\] .After that if it is given that sum of two consecutive odd numbers is \[96\] and we are asked to find the number . so, to solve this we first assume two consecutive odd numbers and according to the question, we equate the addition of two consecutive odd numbers equal to \[96\] and simplify it.

Complete step-by-step answer:
We have to find the odd number before and after \[2n\] . To do so we first need to know what are odd numbers.
Odd numbers are those which are not completely divisible by \[2\] and we also know that between two odd numbers there is an even number and out of which one number is one less than that even number and another number is one more than that of even number.
i.e., if \[x\] is an even number
then \[x - 1\] and \[x + 1\] are odd numbers on each side of it.
So, according to our question \[2n\] is an even number.
\[\therefore 2n - 1\] and \[2n + 1\] are odd numbers on each side of it.
Now, we know that for any two consecutive odd numbers, the difference is \[2\]
And the numbers obtained are also consecutive odd numbers
as \[2n + 1 - (2n - 1) = 2n + 1 - 2n + 1 = 2\]
So, we let the two consecutive numbers as \[2n - 1\] and \[2n + 1\]
And it is given that the sum of two consecutive odd numbers is \[96\]
\[\therefore 2n - 1 + 2n + 1 = 96\]
\[ \Rightarrow 4n = 96\]
Divide both sides by \[4\] we get
\[n = 24\]
\[\therefore \] the consecutive numbers are:
\[2n - 1 = 2(24) - 1 = 47\]
And \[2n + 1 = 2(24) + 1 = 49\]
Hence, the consecutive numbers are \[47\] and \[49\]
So, the correct answer is “Option c”.

Note: In these types of questions, students need to take care while taking the two-consecutive odd numbers. Also, we can apply short methods using the options given in the 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.