# The sum of three consecutive integers is 75, what are the integers?

Jun 3, 2015

Three consecutive integers can be written as $\left(x\right) , \left(x + 1\right) , \left(x + 2\right)$

$\left(x\right) + \left(x + 1\right) + \left(x + 2\right) = 75$
$3 x + 3 = 75$
$3 x = 75 - 3 = 72$
 color(purple)(x = 72/ 3 = 24

so ,$x + 1 = 25 , x + 2 = 26$

the numbers are color(purple)(24,25,26

Jun 7, 2017

This is the type of puzzle that we should try to perform as mental math (in your head). The answer is $24 , 25 , 26.$

#### Explanation:

To arrive at this answer, we can avoid a lot of math formulas, and the use of a calculator by investigating the given statement and quantities.

Three consecutive integers means three numbers that follow each other like $1 , 2 , 3.$ So if we add these up we get $6$ as a sum.

But in this case the desired sum of three consecutive integers is $75$.
That means we can start by dividing the target sum $75$ by $3$.

$\frac{75}{3} = 25 \to$ you have probably memorized that one already.

That means $3 \times 25 = 75 \mathmr{and} 25 + 25 + 25 = 75$

But if we take a $1$ from the first $25$ and move it to the third $25$, we arrive at:

$24 , 25 , 26 \to$ which are three consecutive integers that sum to 75#.

If you can understand this method and are careful in using it you will save a lot of time when these types of puzzles appear.