# 11.6 Applications ii: solving problems  (Page 2/3)

 Page 2 / 3

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.

1. $\begin{array}{cccc}Let& \hfill n& =& \text{the first odd integer. Then,}\hfill \\ & \hfill n+2& =& \text{the second odd integer, and}\hfill \\ & \hfill n+4& =& \text{the third odd integer.}\hfill \end{array}$
2. Translate the words to mathematical symbols and construct an equation. Read phrases.

$\begin{array}{cc}\text{The sum of:}\hfill & \text{add some numbers}\hfill \\ \text{three consecutive odd integers:}\hfill & n,n+2,n+4\hfill \\ \text{is equal to:}\hfill & =\hfill \\ \text{one less than:}\hfill & \text{subtract 1 from}\hfill \\ \text{twice the first odd integer:}\hfill & 2n\hfill \end{array}\right\}n+\left(n+2\right)+\left(n+4\right)=2n-1$

3. Check this result.
The sum of the three integers is

$\begin{array}{ccc}\hfill -7+\left(-5\right)+\left(-3\right)& =& -\text{12}+\left(-3\right)\hfill \\ & =& -\text{15}\hfill \end{array}$

One less than twice the first integer is $2\left(-7\right)-1=-\text{14}-1=-\text{15}$ . Since these two results are equal, the solution checks.

4. The three odd integers are -7, -5, -3.

## Practice set a

When three times a number is decreased by 5, the result is -23. Find the number.

1. Let $x=$
2. Check:
3. The number is .

-6

When five times a number is increased by 7, the result is five less than seven times the number. Find the number.

1. Let $n=$
2. Check:
3. The number is .

6

Two consecutive numbers add to 35. Find the numbers.

1. Check:
2. The numbers are and .

17 and 18

The sum of three consecutive even integers is six more than four times the middle integer. Find the integers.

1. Let $x=$ smallest integer.
= next integer.
= largest integer.
2. Check:
3. The integers are , , and .

-8, -6, and -4

## 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.

1. Let $x=$ the width of the rectangle. Then,
$x+4=$ the length of the rectangle.
2. We can draw a picture.

The length around the rectangle is
$\underset{\text{width}}{\underbrace{x}}+\underset{\text{length}}{\underbrace{\left(x+4\right)}}+\underset{\text{width}}{\underbrace{x}}+\underset{\text{length}}{\underbrace{\left(x+4\right)}}=20$

3. Check:

4. The length of the rectangle is 7 meters.
The width of the rectangle is 3 meters.

## 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.

1. Let $x=$ length of the first leg.
= length of the second leg.
= length of the third leg.
2. We can draw a picture.
3. Check:
4. The lengths of the legs are , , and .

3 inches, 5 inches, and 8 inches

## Exercises

For the following 17 problems, find each solution using the five-step method.

What number decreased by nine is fifteen?

1. Let $n=$ the number.
2. Check:
3. The number is .

24

What number increased by twelve is twenty?

1. Let $n=$ the number.
2. Check:
3. The number is .

If five more than three times a number is thirty-two, what is the number?

1. Let $x=$ the number.
2. Check:
3. The number is .

9

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