Question

Three consecutive integers have a sum of $84$, how do you find the integers?

Answer 1

Hint: The term “consecutive” means one after the other. So we can take the three consecutive integers to be $x,\left( x+1 \right),\left( x+2 \right)$. As stated in the question, we have the three consecutive integers whose sum is equal to $84$. On writing this equation mathematically, we will have a mathematical equation $x+\left( x+1 \right)+\left( x+2 \right)=84$. On solving this equation, we will get the value of $x$. From this value we can determine all of the other integers.

Complete step-by-step answer:
According to the statement given in the question, three consecutive integers have a sum of $84$. Let us assume the three consecutive integers to be $x,\left( x+1 \right),\left( x+2 \right)$. Now, writing the statement of the question mathematically, we obtain the equation
$\Rightarrow x+\left( x+1 \right)+\left( x+2 \right)=84$
Adding all the variable and the constant terms, we have
$\Rightarrow 3x+3=84$
Subtracting $3$ from both the sides of the above equation, we get
$\begin{align}
  & \Rightarrow 3x+3-3=84 \\
 & \Rightarrow 3x=81 \\
\end{align}$
Finally, dividing both the sides of the above equation by $3$ we get
$\begin{align}
  & \Rightarrow \dfrac{3x}{3}=\dfrac{81}{3} \\
 & \Rightarrow x=27 \\
\end{align}$
According to our assumption, the three consecutive integers are $x,\left( x+1 \right),\left( x+2 \right)$. Putting $x=27$ we get the integers as $27,28,29$.
Hence, the integers are $27,28,29$.

Note: We could also assume the three consecutive integers as $\left( x-1 \right),x,\left( x+1 \right)$. By assuming the integers in such a manner, we will be able to obtain a more simple equation containing only the x term on the left hand side of the equation. On adding these integers, we will obtain the equation $3x=84$ which can be solved quickly. Hence, we will be able to obtain the values of the integers more easily and quickly.