



Consecutive integers have the property that if
$\begin{array}{ccc}\hfill n& =& \text{the smallest integer, then}\hfill \\ \hfill n+1& =& \text{the next integer, and}\hfill \\ \hfill n+2& =& \text{the next integer, and so on.}\hfill \end{array}$
Consecutive odd or even integers have the property that if
$\begin{array}{ccc}\hfill n& =& \text{the smallest integer, then}\hfill \\ \hfill n+2& =& \text{the next odd or even integer (since odd or even numbers differ by 2), and}\hfill \\ \hfill n+4& =& \text{the next odd or even integer, and so on.}\hfill \end{array}$
The sum of three consecutive odd integers is equal to one less than twice the first odd integer. Find the three integers.

$\begin{array}{cccc}Let& \hfill n& =& \text{the first odd integer. Then,}\hfill \\ & \hfill n+2& =& \text{the second odd integer, and}\hfill \\ & \hfill n+4& =& \text{the third odd integer.}\hfill \end{array}$
 Translate the words to mathematical symbols and construct an equation. Read phrases.
$\begin{array}{cc}\text{The sum of:}\hfill & \text{add some numbers}\hfill \\ \text{three consecutive odd integers:}\hfill & n,n+\mathrm{2,}n+4\hfill \\ \text{is equal to:}\hfill & =\hfill \\ \text{one less than:}\hfill & \text{subtract 1 from}\hfill \\ \text{twice the first odd integer:}\hfill & 2n\hfill \end{array}\}n+(n+2)+(n+4)=2n1$

$\begin{array}{cc}n+n+2+n+4=2n1\hfill & \\ 3n+6=2n1\hfill & \text{Subtract}2n\text{from}\mathit{\text{both}}\text{sides.}\hfill \\ 3n+62n=2n12n\hfill \\ n+6=1\hfill & \text{Subtract 6 from}\mathit{\text{both}}\text{sides.}\hfill \\ n+66=16\hfill & \\ n=7\hfill & \text{The first integer is 7.}\hfill \\ n+2=7+2=5\hfill & \text{The second integer is 5.}\hfill \\ n+4=7+4=3\hfill & \text{The third integer is 3.}\hfill \end{array}$
 Check this result.
The sum of the three integers is
$\begin{array}{ccc}\hfill 7+(5)+(3)& =& \text{12}+(3)\hfill \\ & =& \text{15}\hfill \end{array}$
One less than twice the first integer is
$2(7)1=\text{14}1=\text{15}$ . Since these two results are equal, the solution checks.
 The three odd integers are 7, 5, 3.
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Practice set a
The sum of three consecutive even integers is six more than four times the middle integer. Find the integers.
 Let
$x=$ smallest integer.
= next integer.
= largest integer.
 Check:
 The integers are
,
, and
.
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Geometry problems
Sample set b
The perimeter (length around) of a rectangle is 20 meters. If the length is 4 meters longer than the width, find the length and width of the rectangle.
 Let
$x=$ the width of the rectangle. Then,
$x+4=$ the length of the rectangle.
 We can draw a picture.
The length around the rectangle is
$\underset{\text{width}}{\underbrace{x}}+\underset{\text{length}}{\underbrace{\left(x+4\right)}}+\underset{\text{width}}{\underbrace{x}}+\underset{\text{length}}{\underbrace{\left(x+4\right)}}=20$

$\begin{array}{cc}x+x+4+x+x+4=\text{20}\hfill & \\ 4x+8=\text{20}\hfill & \text{Subtract 8 from}\mathit{\text{both}}\text{sides.}\hfill \\ 4x=\text{12}\hfill & \text{Divide}\mathit{\text{both}}\text{sides by 4.}\hfill \\ x=3\hfill & \text{Then,}\hfill \\ x+4=3+4=7\hfill & \end{array}$
 Check:
 The length of the rectangle is 7 meters.
The width of the rectangle is 3 meters.
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Practice set b
The perimeter of a triangle is 16 inches. The second leg is 2 inches longer than the first leg, and the third leg is 5 inches longer than the first leg. Find the length of each leg.
 Let
$x=$ length of the first leg.
= length of the second leg.
= length of the third leg.
 We can draw a picture.
 Check:
 The lengths of the legs are
,
, and
.
3 inches, 5 inches, and 8 inches
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Exercises
For the following 17 problems, find each solution using the fivestep method.
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Source:
OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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