Question

Find 3 consecutive positive even integers whose sum is 90

Answer 1

Hint: Here, we will assume the values of the three numbers to be \[2x\], \[2x + 2\] and \[2x + 4\]. We will equate their sum with 90 and form a linear equation. We will solve the linear equation to find the value of \[x\]. Then, we will find the three even numbers by back substituting the value of \[x\].

Complete step-by-step answer:
Let the first even integer be \[{e_1}\], the second even integer be \[{e_2}\] and the third even integer be \[{e_3}\].
We know that an even number is that number which is divisible by 2. So, any even number will be of the form 2 multiplied by another integer. We will assume that:
 \[{e_1} = 2x\]………………………………….\[\left( 1 \right)\]
We know that consecutive even integers differ by 2. So, the second even integer will be the sum of the first even integer and 2:
\[ \Rightarrow {e_2} = 2x + 2\]…………………………………\[\left( 2 \right)\]
The third even integer will be the sum of the second even integer and 2:
\[{e_3} = 2x + 2 + 2\]
\[ \Rightarrow {e_3} = 2x + 4\]…………………………..\[\left( 3 \right)\]
We know that the sum of the three even integers is 90:
\[ \Rightarrow {e_1} + {e_2} + {e_3} = 90\]
We will substitute the values of \[{e_1}\], \[{e_2}\] and \[{e_3}\] from equation (1), equation (2) and equation (3) in the above equation:
\[ \Rightarrow 2x + \left( {2x + 2} \right) + \left( {2x + 4} \right) = 90\]
We will collect the like terms present on the right-hand side of the above equation:
\[ \Rightarrow \left( {2x + 2x + 2x} \right) + \left( {2 + 4} \right) = 90\]
We will add the like terms in the above equation:
\[ \Rightarrow 6x + 6 = 90\]
We will subtract 6 from both sides of the above equation:
\[ \Rightarrow 6x + 6 - 6 = 90 - 6\]
We will simplify the above equation:
\[ \Rightarrow 6x = 84\]
We will divide both sides of the above equation by 6:
\[ \Rightarrow \dfrac{{6x}}{6} = \dfrac{{84}}{6}\]
We will further simplify the above equation:
\[ \Rightarrow x = 14\]
Substituting 14 for \[x\] in equation \[\left( 3 \right)\], we get
\[\begin{array}{l}{e_1} = 2\left( {14} \right)\\ \Rightarrow {e_1} = 28\end{array}\]
Substituting 14 for \[x\] in equation \[\left( 2 \right)\], we get
\[\begin{array}{l}{e_2} = 2\left( {14} \right) + 2\\ \Rightarrow {e_2} = 30\end{array}\]
Substituting 14 for \[x\] in equation \[\left( 3 \right)\], we get
\[\begin{array}{l}{e_3} = 2\left( {14} \right) + 4\\ \Rightarrow {e_3} = 32\end{array}\]
$\therefore $ The 3 consecutive positive even integers whose sum is 90 are 28, 30 and 32.

Note: Even numbers are the numbers that are divided by 2 whereas the numbers that are not divisible by 2 are called odd numbers. Just like even numbers, consecutive odd numbers also differ by 2. Odd numbers are those numbers that aren’t even; that is, they are the numbers that are not divisible by 2. They leave a remainder of 1 on division with 2.