The sum of three consecutive integers is 1,623. What are the numbers?

Sep 18, 2016

The three consecutive integers are $540 , 541 , 542$.

Explanation:

Three consecutive integers are three numbers in a row. For example, 4, 5 and 6 are three consecutive integers. If you start with the first number, you get the second number by adding 1 to the first number (4+1=5). You get the third number by adding 2 to the first number (4+2=6).

Let's call the first number (integer) $\textcolor{b l u e}{x}$.

Find the second number by adding 1 to the first. So the
2nd consecutive integer is $\textcolor{red}{x + 1}$

Find the 3rd number by adding 2 to the first. The 3rd consecutive integer is $\textcolor{\lim e g r e e n}{x + 2}$.

The problem also states that the sum of the three consecutive integers is $1 , 623$. The word "sum" means the answer to an addition problem. So, we add the three numbers and set them equal to $\textcolor{m a \ge n t a}{1 , 623}$.

$\textcolor{b l u e}{x} \textcolor{w h i t e}{a a} + \textcolor{red}{x + 1} \textcolor{w h i t e}{a a} + \textcolor{\lim e g r e e n}{x + 2} = \textcolor{m a \ge n t a}{1623}$

Combine like terms. First, add the three x's.

$3 x + 1 + 2 = 1623$

Next, add the 1 and the 2.

$3 x + 3 = 1623$
$\textcolor{w h i t e}{a}$
$\textcolor{w h i t e}{a a} - 3 \textcolor{w h i t e}{a a a} - 3 \textcolor{w h i t e}{a a a a}$Subtract 3 from both sides.

$3 x = 1620$

Divide both side by 3.

$\frac{3 x}{3} = \frac{1620}{3}$

$\textcolor{b l u e}{x} = 540$

The first consecutive integer is $\textcolor{b l u e}{540}$.

Find the 2nd number by adding 1 to the first.
The 2nd consecutive integer is $540 + 1 = \textcolor{red}{541}$

Find the 3rd number by adding 2 to the first.
The 3rd consecutive integer is $540 + 2 = \textcolor{\lim e g r e e n}{542}$

These three number "in a row" are three consecutive integers. Their sum is 1623. Let's check:

$\textcolor{b l u e}{540} + \textcolor{red}{541} + \textcolor{\lim e g r e e n}{542} = \textcolor{m a \ge n t a}{1623}$