Question

Find three consecutive odd integers such that the sum of first and third integers is the same as the second integer when decreased by $9$.
A. $ - 9, - 7, - 5$
B. $ - 13, - 11, - 9$
C. \[ - 15, - 13, - 11\]
D. $ - 11, - 9, - 7$

Answer 1

Hint: As we know sum is the result of adding two or more numbers and the term decreased by represents that a quantity is decreased by some specific units. We will use this information to form algebraic equations in one variable.

Complete step by step solution:
We have to consider random variables for the unknown terms in the question. As we are asked to find the three consecutive odd integers and we don’t know them we will take the first number to be a variable and then try to find its value. Let \[x\] be the smallest and first odd integer. Now, since we know that the three numbers are consecutive odd integers and the difference between any two such integers is two, we can observe that the required integers are $x,x + 2,x + 4$.
It is given that the sum of the first and third integer is the same as the second integer decreased by 9. That is, the sum of $x$ and $x + 4$ is equal to the second integer $x + 2$ decreased by 9.
We can phrase this stamen as an algebraic equation. Remember that sum means addition of two numbers and decreased by 9 implies subtracting 9 from the quantity. So, we get the algebraic equation as $x + (x + 4) = (x + 2) - 9$
$ \Rightarrow 2x + 4 = x - 7$
$ \Rightarrow 2x - x = - 7 - 4$
$ \Rightarrow x = - 11$
Therefore, the required consecutive odd integers are $ - 11, - 9, - 7$.
Hence, option (D) $ - 11, - 9, - 7$ is the correct answer.

Note:
Remember that since we are given a relationship between the three unknowns, we wrote the algebraic equation in terms of only one variable. But if we are not given about the relationship between them, we have to take three variables each representing a separate integer and form the equation. And in such a case to find them we need three equations containing the variables.