Question

Three consecutive integers have a sum of $78,$ how do you find the integers.

Answer 1

Hint: As here we have to determine the three consecutive numbers, so, for that let the consecutive number be\[x\], $''x+1''$ and $''x+2''$, then solve the obtained equation for determining the value of smallest number, and then determine the value of second and third number.

Complete step-by-step solution:
As per data given in the question,
We have to find the value of three numbers which are consecutive and their sum is $78.$
As we know that,
Consecutive means in a continuous manner, like $1,2,3....$ are consecutive numbers.
So,
Let the first consecutive number is $=x$
So, the next consecutive integer will be 1 more than the value of \[x\]
So, value of second integer will be $=\left( x+1 \right)$
So, we can say that,
The value of third consecutive number will be one more than the value of second number,
So, it will be $=\left( x+2 \right)$
Hence,
As we have,
That the sum of all three number is $78$
So,
We can say that,
$\left( x+x+1+x+2 \right)=78$
Now for determining the value of \[x\] or the value of the smallest integer we need to solve the equation.
So, for solving the equation,
Let’s shift the like terms at one side,
So, we will get,
$\Rightarrow 3x+3=78$
$\Rightarrow 3x=78-3$
As, here 3 is in positive in left side, so when we shift $3$ from left to right side of equation its sign gets reversed
So,
$\Rightarrow 3x=75$
Now, as here 3 is in multiplication with \[x\] so when we shift 3 from left to right side of the equation it will shift in the division.
So,
$\Rightarrow 3x=75$
Hence, value of \[x\] will be $=\dfrac{75}{3}=25$
So,
We can say that,
The minimum value of integer will be $25$
So, next number will be $\left( x+1 \right)=25+1=26$
So, third number will be $=\left( x+2 \right)=25+2=27$

The value of consecutive integer will be $25,26$ and $27.$

Additional Information: Integers are such numbers that are written without a fractional value.
Integers are basically of two types:
One is positive integer and the other one is negative integer.
Positive integers start from $+1$ and end up to infinity, while the negative integers start from $-1$ and end at minus infinity.
Zero, is neither positive integer nor negative integer.
Integers are little different from the real numbers,
As real numbers are those numbers which starts from zero and ends up to infinity,
Means, we can say that, if we add negative integers in real numbers it will show the group of the integers.
Points to remember:
All natural numbers are real numbers, but the vice versa is not correct, as zero is a real number but it is not a natural number.
All real numbers are integers but the vice versa is not correct.

Note: Always consider three numbers as \[x\], $''x+1''$ and $''x+2'',$ as it is said that all the required numbers are consecutive.
Don’t take numbers as \[“x”\], $''y''$ and $''z''$ as here only one equation is formed with the help of data given in the question, as we know that we need a minimum of three equations to determine the value of three variables.