To solve the problem, let's start by defining what we're working with. Consecutive integers are numbers that follow each other in order, with no gaps between them. For example, if you pick any number \( n \), the next consecutive integer would be \( n+1 \), and the one after that would be \( n+2 \).
The problem states that the sum of three consecutive integers is 90. Let's define our integers:
Let \( x \) be the first integer.
Then \( x + 1 \) would be the second consecutive integer.
And \( x + 2 \) would be the third consecutive integer.
According to the problem, the sum of these three consecutive integers equals 90. We can write this as an equation:
\[ x + (x + 1) + (x + 2) = 90 \]
Now, let's solve the equation step by step:
Combine like terms (collect all terms with \( x \) together):
\[ x + x + x + 1 + 2 = 90 \]
\[ 3x + 3 = 90 \]
Now let's isolate \( x \) by first subtracting 3 from both sides of the equation:
\[ 3x = 90 - 3 \]
\[ 3x = 87 \]
Next, divide both sides of the equation by 3 to find the value of \( x \):
\[ x = \frac{87}{3} \]
\[ x = 29 \]
So, the first integer is 29.
Given that the integers are consecutive, the second integer is one more than the first:
\[ x + 1 = 29 + 1 \]
\[ x + 1 = 30 \]
The second integer is 30.
The third integer is two more than the first (or one more than the second):
\[ x + 2 = 29 + 2 \]
\[ x + 2 = 31 \]
The third integer is 31.
Therefore, the three consecutive integers whose sum is 90 are 29, 30, and 31.
