Lesson Problems with consecutive integer numbers; odd even consecutive integer numbers

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<H2>Problems with consecutive integer numbers; odd/even consecutive integer numbers</H2>

<H3>Problem 1</H3>The sum of three consecutive integers is &nbsp;240. &nbsp;Find the integers.

<B>Solution 1</B>

This is a very simple problem.
Let &nbsp;{{{x}}}&nbsp; be the second &nbsp;(the middle)&nbsp; of these three consecutive integers.
Then the first one is &nbsp;{{{x-1}}}, &nbsp;while the third one is &nbsp;{{{x+1}}}.
Since the sum of these three consecutive integers is equal to &nbsp;240, &nbsp;you can write the equation

{{{(x-1) + x + (x+1) = 240}}}.

Simplify this equation. &nbsp;You will get 
{{{3x = 240}}} &nbsp;&nbsp;(after combining like terms at the left side),
{{{x = 80}}}   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(after dividing both sides by 3).

Thus, &nbsp;the three consecutive integers are &nbsp;79, &nbsp;80 &nbsp;and &nbsp;81.
You can easily &nbsp;<B>check</B>&nbsp; this solution: 
{{{79 + 80 + 81 = 240}}}.

<B>Solution 2</B>

This time let &nbsp;{{{x}}}&nbsp; be the first &nbsp;(the smallest)&nbsp; of these three consecutive integers.
Then the second one is &nbsp;{{{x+1}}}, &nbsp;while the third one is &nbsp;{{{x+2}}}.
Since the sum of these three consecutive integers is equal to &nbsp;240, &nbsp;you can write the equation

{{{x + (x+1) + (x+2) = 240}}}.

Simplify this equation step by step. &nbsp;You will get 
{{{3x + 3 = 240}}} &nbsp;&nbsp;(after combining like terms at the left side),
{{{3x = 237}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(after moving the constant term &nbsp;<B>3</B>&nbsp; to the right side with the opposite sign and combining like terms at the right side),  
{{{x = 79}}}   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(after dividing both sides by &nbsp;3).

Thus, &nbsp;the solution is the same: &nbsp;the three consecutive integers are &nbsp;79, &nbsp;80 &nbsp;and &nbsp;81.

<B>Answer</B>. &nbsp;The three consecutive integers are &nbsp;79, &nbsp;80 &nbsp;and &nbsp;81.

<H3>Problem 2</H3>One third of the sum of five consecutive integers is &nbsp;15. &nbsp;Find the integers.

<B>Solution</B>

Again, &nbsp;this is a simple problem.
Since one third of the sum of five consecutive integers is &nbsp;15, &nbsp;the entire sum of five consecutive integers is &nbsp;45.
Let &nbsp;{{{x}}}&nbsp; be the third &nbsp;(the middle)&nbsp; of these five consecutive integers.
Then the first integer is equal to &nbsp;{{{x-2}}},
the second integer is equal to &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x-1}}},
the fourth integer is equal to &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x+1}}}, &nbsp;and
the fifth integer is equal to &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x+2}}}.

Since the sum of these five consecutive integers is equal to &nbsp;45, &nbsp;you can write the equation

{{{(x-2) + (x-1) + x + (x+1) + (x+2) = 45}}}.

Simplify this equation. &nbsp;You will get 
{{{5x = 45}}} &nbsp;&nbsp;(after combining like terms at the left side),
{{{x = 9}}}   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(after dividing both sides by &nbsp;5).

Thus, &nbsp;the five consecutive integers are &nbsp;7, &nbsp;8, &nbsp;9, &nbsp;10 &nbsp;and &nbsp;11.

You can easily &nbsp;<B>check</B>&nbsp; that
{{{7 +  9 + 10 + 11 = 45}}}, 
hence one third of the sum is &nbsp;9. &nbsp;The solution is correct.

<B>Answer</B>. &nbsp;The five consecutive integers are &nbsp;7, &nbsp;8, &nbsp;9, &nbsp;10 &nbsp;and &nbsp;11.

<H3>Problem 3</H3>The sum of three consecutive even integers is &nbsp;18. &nbsp;Find the integers.

<B>Solution</B>

Let &nbsp;{{{x}}}&nbsp; be the second &nbsp;(the middle)&nbsp; of these three consecutive even integers.
Then the first one is &nbsp;{{{x-2}}}, &nbsp;while the third one is &nbsp;{{{x+2}}}.
Since the sum of these three consecutive even integers &nbsp;{{{x-2}}}, &nbsp;{{{x}}} &nbsp;and {{{x+2}}}&nbsp; is equal to &nbsp;18, &nbsp;you can write the equation

{{{(x-2) + x + (x+2) = 18}}}.

Simplify this equation. &nbsp;You will get 
{{{3x = 18}}} &nbsp;&nbsp;(after combining like terms at the left side),
{{{x = 6}}}   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(after dividing both sides by &nbsp;3).

Thus, &nbsp;the three consecutive even integers are &nbsp;6-2=4, &nbsp;6, &nbsp;and &nbsp;6+2=8.
Easy &nbsp;<B>check</B>&nbsp; shows that the solution is correct:
{{{4 + 6 + 8 = 18}}}.

<B>Answer</B>. &nbsp;The three consecutive even integers are &nbsp;4, &nbsp;6 and &nbsp;8.

<H3>Problem 4</H3>The sum of three consecutive odd integers is &nbsp;27. &nbsp;Find the integers.

<B>Solution</B>

Let &nbsp;{{{x}}}&nbsp; be the second &nbsp;(the middle)&nbsp; of these three consecutive odd integers.
Then the first one is &nbsp;{{{x-2}}}, &nbsp;while the third one is &nbsp;{{{x+2}}}.
Since the sum of these three consecutive odd integers is equal to &nbsp;27, &nbsp;you can write the equation

{{{(x-2) + x + (x+2) = 27}}}.

Simplify this equation. &nbsp;You will get 
{{{3x = 27}}} &nbsp;&nbsp;(after combining like terms at the left side),
{{{x = 9}}}   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(after dividing both sides by 3).

Thus, &nbsp;the three consecutive odd integers are &nbsp;9-2=7, &nbsp;9, &nbsp;and &nbsp;9+2=11.
Easy &nbsp;<B>check</B>&nbsp; shows that the solution is correct:
{{{7 + 9 + 11 = 27}}}.

<B>Answer</B>. &nbsp;The three consecutive odd integers are &nbsp;7, &nbsp;9 &nbsp;and &nbsp;11.

<H3>Problem 5</H3>The sum of four consecutive odd integers is &nbsp;40. &nbsp;Find the integers.

<B>Solution</B>

This time there is no the unique middle term of four consecutive odd integers.
Therefore let us introduce the unknown &nbsp;{{{x}}}&nbsp; as the first &nbsp;(minimal)&nbsp; of these four consecutive odd integers.
Then the second number is &nbsp;{{{x+2}}}, &nbsp;the third one is &nbsp;{{{x+4}}} &nbsp;and the fourth one is &nbsp;{{{x+6}}}.
Since the sum of these four consecutive odd integers is equal to &nbsp;40, &nbsp;you can write the equation

{{{x + (x+2) + (x+4) + (x+6)= 40}}}.

Simplify this equation. &nbsp;You will get 
{{{4x + 12 = 40}}} &nbsp;&nbsp;(after combining like terms at the left side),
{{{4x = 28}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(after moving the constant term &nbsp;<B>12</B>&nbsp; to the right side with the opposite sign and combining like terms at the right side),  
{{{x = 7}}}   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(after dividing both sides by &nbsp;4).

Thus, &nbsp;the three consecutive odd integers are &nbsp;7, &nbsp;7+2=9, &nbsp;9+2=11, &nbsp;and &nbsp;11+2=13.
Easy &nbsp;<B>check</B>&nbsp; shows that the solution is correct:
{{{7 + 9 + 11 + 13 = 40}}}.

<B>Answer</B>. &nbsp;The four consecutive odd integers are &nbsp;7, &nbsp;9, &nbsp;11 &nbsp;and &nbsp;13.