This is the first of the Solving Consecutive Number Series, a series of post discussing word problems about consecutive numbers.
Consecutive numbers are numbers that follow each other in order. In number problems in Algebra, consecutive numbers usually have difference 1 or 2. Below are the types of consecutive numbers,
consecutive numbers – 5, 6, 7, 8, …
consecutive even numbers – 16, 18, 20, 22…
consecutive odd numbers – 3, 5, 7, 8, …
The symbol … means that the list may be continued.
Notice that consecutive numbers always increase by 1 in each term. If we make 5 as point of reference, then, we can write the numbers above as
5, 5 + 1, 5 + 2, 5 + 3.
That means that if our first number is x, then the list above can be written as
x, (x + 1), (x + 2), (x + 3)
and so on.
As for the consecutive even and consecutive odd numbers above, with the smallest numbers as point of reference, they can be written as
16, 16 + 2, 16 + 4, 16 + 6.
3, 3 + 2, 3 + 4, 3 + 6.
Notice that both consecutive odd and consecutive integers increase by 2 in each time. So, if we let x be the first number, the terms can be written as
x, (x + 2), (x + 4), (x + 6)
and so on.
Now, that we know how to represent consecutive numbers, let us solve our first problem.
The sum of two consecutive numbers is 81. What are the numbers?
Since there is no mention of odd or even, the terms only increase by 1. So, let
x = the first number
x + 1 = the second number.
The word sum means we have to add and the phrase “is 81” means that we have to equate the sum to 81. That is
first number + second number = 81.
Since the first number is x and the second number is (x + 1),
x + x + 1 = 81.
Solving the equation, we have
2x + 1 = 81.
Subtracting 1 from both sides, we have
2x = 80.
Dividing both sides by 2 results to
x = 40.
So, the smaller number is 40, and the larger number is 40 + 1 = 41. The consecutive numbers are 40, 41 and their sum is 81.
The sum of three consecutive even numbers is 42. What are the numbers?
In this example, we have 3 consecutive even numbers. Recall that form above, consecutive even numbers increase by 2 each time. So, let
x = first number
x + 2 = second number
x + 4 = second number.
Again, the problem mentioned the word sum, so we have to add. That is,
first number + second number + third number = 42.
Substituting the algebraic representation above, we have
x + (x + 2) + (x + 4) = 42
Solving the equation,
3x + 6 = 42.
Subtracting 6 from both sides, we have
3x = 36.
Dividing both sides of the equation by 3 results to
x = 12.
So, 12 is the smallest number, 12 + 2 = 14 is the second number and 12 + 4 = 16 is the largest number. The consecutive numbers are 12, 14, and 16 and their sum is 42.
You can watch the explanation of this article in the video below. The language is Taglish.
In the next post, we will be discussing more consecutive number problems.