Question

The sum of four consecutive integers is $2174$. How do you find the numbers?

Answer 1

Hint:Here in this question, first of all we assume four consecutive integers as $\left( x \right)$,$\left( {x + 1} \right)$,$\left( {x + 2} \right)$ and $\left( {x + 3} \right)$.Next, we add all these four integers and equate them to $2174$. Then, we find the value of $x$ from the obtained equation, then from $x$, we find the value of other consecutive integers.

Complete step by step answer:
Here, in this question we are given that the sum of four consecutive integers is $2174$ and we are supposed to find the integers.Let us assume that our first integer is $\left( x \right)$.We do know that the consecutive numbers differ by $1$. So, let the next consecutive integer be $\left( {x + 1} \right)$. Now, again the next consecutive integer would differ by $1$ from the previous integer. So, we take the third integer as $\left( {x + 2} \right)$ and the fourth integer as $\left( {x + 3} \right)$. Therefore, we get,
Value of the first integer =$\left( x \right)$----(1)
Value of the second integer =$\left( {x + 1} \right)$----(2)
Value of the third integer =$\left( {x + 2} \right)$-----(3)
Value of the fourth integer =$\left( {x + 3} \right)$-----(4)

Given is that, the sum of four consecutive integers is $2174$. So, we get,
(First Integer) + (Second Integer) + (Third Integer) + (Fourth Integer) =$2174$
By substituting the values of first, second, third and fourth integers from equation (1), (2), (3) and (4) respectively, we get,
\[\left( x \right) + \left( {x + 1} \right) + \left( {x + 2} \right) + \left( {x + 3} \right) = 2174\]
Rearrange the terms of the above obtained equation so that all the like terms are together.
\[\left( {x + x + x + x} \right) + \left( {1 + 2 + 3} \right) = 2174 \\
\Rightarrow 4x + 6 = 2174 \\ \]
Subtract\[6\] from both sides of equation and we get,
\[4x + 6 - 6 = 2174 - 6 \\
\Rightarrow 4x = 2168 \\ \]
Divide both the sides of the equation by\[4\]and we get,
\[\dfrac{{4x}}{4} = \dfrac{{2168}}{4} \\
\Rightarrow x = 542 \\ \]
So, we get the value of our first integer =\[\left( x \right) = 542\]
Value of second integer =\[\left( {x + 1} \right) = 543\]
Value of third integer =\[\left( {x + 2} \right) = 544\]
Value of fourth integer =\[\left( {x + 3} \right) = 545\]

Hence, the four consecutive integers are \[542,543,544,545\].

Note: Students can double check their answer by adding \[542,543,544\] and \[545\]. If the value of the sum is equal to\[2174\] then, their solution is correct. One common mistake which usually happens in these types of questions is that sometimes students take four consecutive numbers as \[\left( {x + 1} \right)\],\[\left( {x + 2} \right)\],\[\left( {x + 3} \right)\]and\[\left( {x + 4} \right)\] by doing so they get wrong answer.