Question

Three consecutive integers add up to $51$. What are these integers?
A. $15,15,12$
B. $16,17,18$
C. $10,17,22$
D. $16,17,42$

Answer 1

Hint: We will assume the three integers to be added up to $51$ as $x,y,z$. In the problem they have mentioned that the three integers are consecutive integers i.e. the next element in the series is formed by adding one to the before/previous element. From this statement we can have the values of $y,z$ in terms of $x$. Now all the three variables are in terms of $x$, and then we will sum them and equate to $51$ as mentioned in the problem, then we will get the value of $x$. From the values of $x$, then we will calculate the values of other two integers by using the value of $x$.

Complete step-by-step answer:
Let the three integers are $x,y,z$.
Given that the three integers are consecutive, then we will have
$y=x+1$, $z=y+1$
The sum of the three integers is given by
$S=x+y+z$
Substituting the values of $y,z$ in the above equation, then we will get
$S=x+\left( x+1 \right)+\left( y+1 \right)$
Simplifying the above equation and substituting the value $y=x+1$ in the above equation, then we will get
$\begin{align}
  & S=x+x+1+\left( \left( x+1 \right)+1 \right) \\
 & =2x+1+x+1+1 \\
 & =3x+3 \\
\end{align}$
In the problem they have mentioned the sum of the three integers is $51$. So, by equating the value $S$ to the given value we will get
$S=51$
Substituting the value of $S=3x+3$ in the above equation, then we will get
$3x+3=51$
Subtracting $3$ from both side of the above equation, then we will have
$3x+3-3=51-3$
We know that $a-a=0$, then
$3x=48$
Now dividing the both sides of above equation with $3$, then we will get
$\begin{align}
  & \dfrac{3x}{3}=\dfrac{48}{3} \\
 & x=16 \\
\end{align}$
Here we got the value of $x$ as $16$.
Hence the first integer is $16$, second consecutive integer is $16+1=17$, third consecutive integer is $17+1=18$.

So, the correct answer is “Option (c)”.

Note: For this problem we can directly observe the given options and tick the right option. In the problem they have mentioned consecutive integers, so find whether the options have the consecutive integers or not. Luckily here we have only one option that has consecutive integers, so we can directly tick that option as the correct option.