11.6: Applications II- Solving Problems

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Learning Objectives

be more familiar with the five-step method for solving applied problems
be able to use the five-step method to solve number problems and geometry problems

The Five Step Method

We are now in a position to solve some applied problems using algebraic methods. The problems we shall solve are intended as logic developers. Although they may not seem to reflect real situations, they do serve as a basis for solving more complex, real situation, applied problems. To solve problems algebraically, we will use the five-step method.

Strategy for Reading Word Problems
When solving mathematical word problems, you may wish to apply the following "reading strategy." Read the problem quickly to get a feel for the situation. Do not pay close attention to details. At the first reading, too much attention to details may be overwhelming and lead to confusion and discouragement. After the first, brief reading, read the problem carefully in phrases. Reading phrases introduces information more slowly and allows us to absorb and put together important information. We can look for the unknown quantity by reading one phrase at a time.

Five-Step Method for Solving Word Problems

Let $x$ (or some other letter) represent the unknown quantity.
Translate the words to mathematical symbols and form an equation. Draw a picture if possible.
Solve the equation.
Check the solution by substituting the result into the original statement, not equation, of the problem.
Write a conclusion.

If it has been your experience that word problems are difficult, then follow the five-step method carefully. Most people have trouble with word problems for two reasons:

They are not able to translate the words to mathematical symbols. (See [link].)
They neglect step 1. After working through the problem phrase by phrase, to become familiar with the situation,

INTRODUCE A VARIABLE

Number Problems

Sample Set A

What number decreased by six is five?

Solution

Let $n$ represent the unknown number.
Translate the words to mathematical symbols and construct an equation. Read phrases.
$\left \{ \begin{array} {lr} {\text{What number: }} & {n} \\ {\text{decreased by:}} & {-} \\ {\text{six:}} & {6} \\ {\text{is:}} & {=} \\ {\text{five:}} & {5} \end{array} \right \} n - 6 = 5$
Solve the equation.
$n - 6 = 5$ Add 6 to both sides.
$n - 6 + 6 = 5 + 6$
$n = 11$
Check the result.
When 11 is decreased by 6, the result is $11 - 6$, which is equal to 5. The solution checks.
The number is 11.

Sample Set A

When three times a number is increased by four, the result is eight more than five times the number.

Solution

Let $x =$ the unknown number.
Translate the phrases to mathematical symbols and construct an equation.
$\left \{ \begin{array} {lr} {\text{When three times a number: }} & {3x} \\ {\text{is increased by:}} & {+} \\ {\text{four:}} & {4} \\ {\text{the result is:}} & {=} \\ {\text{eight:}} & {8} \\ {\text{more than:}} & {+} \\ {\text{five times the number:}} & {5x} \end{array} \right \} 3x + 4 = 5x + 8$
$\begin{array} {ll} {3x + 4 = 5x + 8} & {\text{Subtract 3x from } both \text{ sides.}} \\ {3x + 4 - 3x = 5x + 8 - 3x} & {} \\ {4 = 2x + 8} & {\text{Subtract 8 from } both \text{ sides}} \\ {4 - 8 = 2x + 8 - 8} & {} \\ {-4 = 2x} & {\text{Divide } both \text{ sides by 2.}} \\ {-2 = x} & {} \end{array}$
Check the result.
Three times -2 is -6. Increasing -6 by 4 results in -6 + 4 = -2. Now, five times -2 is -10.
Increasing -10 by 8 results in -10 + 8 = -2. The results agree, and the solution checks.
The number is -2.

Sample Set A

Consecutive integers have the property that if

$\begin{array} {rcl} {n} & = & {\text{the smallest integer, then}} \\ {n + 1} & = & {\text{the next integer, and}} \\ {n + 2} & = & {\text{the next integer, and so on.}} \end{array}$

Consecutive odd or even integers have the property that if

$\begin{array} {rcl} {n} & = & {\text{the smallest integer, then}} \\ {n + 2} & = & {\text{the next odd or even integer (since odd or even numbers differ by 2), and}} \\ {n + 4} & = & {\text{the next odd or even integer, and so on.}} \end{array}$

The sum of three consecutive odd integers is equal to one less than twice the first odd integer. Find the three integers.

Solution

Let $\begin{array} {rcl} {n} & = & {\text{the first odd integer. Then,}} \\ {n + 2} & = & {\text{the second odd integer, and}} \\ {n + 4} & = & {\text{the third odd integer.}} \end{array}$
Translate the words to mathematical symbols and construct an equation. Read phrases.
$\left \{ \begin{array} {lr} {\text{The sum of: }} & {\text{add some numbers}} \\ {\text{three consecutive odd integers:}} & {n, n + 2, n + 4} \\ {\text{is equal to:}} & {=} \\ {\text{one less than:}} & {\text{subtract 1 from}} \\ {\text{twice the first odd integer:}} & {2n} \end{array} \right \} n + (n + 2) + (n + 4) = 2n - 1$
$\begin{array} {ll} {n + n + 2 + n + 4 = 2n - 1} & {} \\ {3n + 6 = 2n - 1} & {\text{Subtract } 2n \text{ from } both \text{ sides.}} \\ {3n + 6 - 2n = 2n - 1 - 2n} & {} \\ {n + 6 = -1} & {\text{Subtract 6 from } both \text{ sides}} \\ {n + 6 - 6 = -1 -6} & {} \\ {n = -7} & {\text{The first integer is -7.}} \\ {n + 2 = -7 + 2 = -5} & {\text{The second integer is -5.}} \\ {n + 4 = -7 + 4 = -3} & {\text{The third integer is -3.}} \end{array}$
Check this result.
The sum of the three integers is
$\begin{array} {rcl} {-7 + (-5) + (-3)} & = & {-12 + (-3)} \\ {} & = & {-15} \end{array}$
One less than twice the first integer is $2(-7) - 1 = -14 - 1 = -15$. Since these two results are equal, the solution checks.
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.

Let $x =$
Check:
The number is .

Answer: -6

Practice Set A

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

Let $n =$
Check:
The number is .

Answer: 6

Practice Set A

Two consecutive numbers add to 35. Find the numbers.

Check:
The numbers are and .

Answer: 17 and 18

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 number are , . and .

Answer: -8, -6 and -4

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.

Solution

Let $x =$ the width of the rectangle. Then,
$x + 4 =$ the length of the rectangle.
We can draw a pricture.
$A rectangle with base length the quantity x + 4 and height x.$
The length around the rectangle is
$\underbrace{x}_{\text{width}} + \underbrace{(x + 4)}_{\text{length}} + \underbrace{x}_{\text{width}} + \underbrace{(x + 4)}_{\text{length}} = 20$
$\begin{array} {ll} {x + x + 4 + x + x + 4 = 20} & {} \\ {4x + 8 = 20} & {\text{Subtract 8 from } both \text{ sides.}} \\ {4x = 12} & {\text{Divide } both \text{ sides by 4.}} \\ {x = 3} & {\text{Then,}} \\ {x + 4 = 3 + 4 = 7} & {} \end{array}$
Check:
$A rectangle with base length 7 and height 3.$
$graphics3.png$
The length of the rectangle is 7 meters.
The width of the rectangle is 3 meters.

Pracitce 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 .

Answer: 3 inches, 5 inches, and 8 inches

Exercises

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

Exercise $\PageIndex{1}$

What number decreased by nine is fifteen?

Let $n =$ the number.
Check:
The number is .

Answer: 24

Exercise $\PageIndex{2}$

What number increased by twelve is twenty?

Let $n =$ the number.
Check:
The number is .

Exercise $\PageIndex{3}$

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

Let $x =$ the number.
Check:
The number is .

Answer: 9

Exercise $\PageIndex{4}$

If four times a number is increased by fifteen, the result is five. What is the number?

Let $x =$
Check:
The number is .

Exercise $\PageIndex{5}$

When three times a quantity is decreased by five times the quantity, the result is negative twenty. What is the quantity?

Let $x =$
Check:
The quantity is .

Answer: 10

Exercise $\PageIndex{6}$

If four times a quantity is decreased by nine times the quantity, the result is ten. What is the quantity?

Let $y = $
Check:
The quantity is .

Exercise $\PageIndex{7}$

When five is added to three times some number, the result is equal to five times the number decreased by seven. What is the number?

Let $n =$
Check:
The number is .

Answer: 6

Exercise $\PageIndex{8}$

When six times a quantity is decreased by two, the result is six more than seven times the quantity. What is the quantity?

Let $x =$
Check:
The quantity is .

Exercise $\PageIndex{9}$

When four is decreased by three times some number, the result is equal to one less than twice the number. What is the number?

Check:

Answer: 1

Exercise $\PageIndex{10}$

When twice a number is subtracted from one, the result is equal to twenty-one more than the number. What is the number?

Exercise $\PageIndex{11}$

The perimeter of a rectangle is 36 inches. If the length of the rectangle is 6 inches more than the width, find the length and width of the rectangle.

Let $w =$ the width.
= the length.
We can draw a picture.
$A rectangle.$
Check:
The length of the rectangle is inches, and the width is inches.

Answer: Length=12 inches, Width=6 inches

Exercise $\PageIndex{12}$

The perimeter of a rectangle is 48 feet. Find the length and the width of the rectangle if the length is 8 feet more than the width.

Let $w =$ the width.
= the length.
We can draw a picture.
$A rectangle.$
Check:
The length of the rectangle is feet, and the width is feet.

Exercise $\PageIndex{13}$

The sum of three consecutive integers is 48. What are they?

Let $n =$ the smallest integer.
= the next integer
= the next integer
Check:
The three integers are , , and .

Answer: 15, 16, 17

Exercise $\PageIndex{14}$

The sum of three consecutive integers is -27. What are they?

Let $n =$ the smallest number.
= the next integer.
= the next integer.
Check:
The three integers are , , and .

Exercise $\PageIndex{15}$

The sum of five consecutive integers is zero. What are they?

Let $n =$
The five integers are , , , , and .

Answer: -2, -1, 0, 1, 2

Exercise $\PageIndex{16}$

The sum of five consecutive integers is -5. What are they?

Let $n =$
The five integers are , , , , and .

Continue using the five-step procedure to find the solutions.

Exercise $\PageIndex{17}$

The perimeter of a rectangle is 18 meters. Find the length and width of the rectangle if the length is 1 meter more than three times the width.

Answer: Length is 7, width is 2

Exercise $\PageIndex{18}$

The perimeter of a rectangle is 80 centimeters. Find the length and width of the rectangle if the length is 2 meters less than five times the width.

Exercise $\PageIndex{19}$

Find the length and width of a rectangle with perimeter 74 inches, if the width of the rectangle is 8 inches less than twice the length.

Answer: Length is 15, width is 22

Exercise $\PageIndex{20}$

Find the length and width of a rectangle with perimeter 18 feet, if the width of the rectangle is 7 feet less than three times the length.

Exercise $\PageIndex{21}$

A person makes a mistake when copying information regarding a particular rectangle. The copied information is as follows: The length of a rectangle is 5 inches less than two times the width. The perimeter of the rectangle is 2 inches. What is the mistake?

Answer: The perimeter is 20 inches. Other answers are possible. For example, perimeters such as 26, 32 are possible.

Exercise $\PageIndex{22}$

A person makes a mistake when copying information regarding a particular triangle. The copied information is as follows: Two sides of a triangle are the same length. The third side is 10 feet less than three times the length of one of the other sides. The perimeter of the triangle is 5 feet. What is the mistake?

Exercise $\PageIndex{23}$

The perimeter of a triangle is 75 meters. If each of two legs is exactly twice the length of the shortest leg, how long is the shortest leg?

Answer: 15 meters

Exercise $\PageIndex{24}$

If five is subtracted from four times some number the result is negative twenty-nine. What is the number?

Exercise $\PageIndex{25}$

If two is subtracted from ten times some number, the result is negative two. What is the number?

Answer: $n = 0$

Exercise $\PageIndex{26}$

If three less than six times a number is equal to five times the number minus three, what is the number?

Exercise $\PageIndex{27}$

If one is added to negative four times a number the result is equal to eight less than five times the number. What is the number?

Answer: $n = 1$

Exercise $\PageIndex{28}$

Find three consecutive integers that add to -57.

Exercise $\PageIndex{29}$

Find four consecutive integers that add to negative two.

Answer: -2, -1, 0, 1

Exercise $\PageIndex{30}$

Find three consecutive even integers that add to -24.

Exercise $\PageIndex{31}$

Find three consecutive odd integers that add to -99.

Answer: -35, -33, -31

Exercise $\PageIndex{32}$

Suppose someone wants to find three consecutive odd integers that add to 120. Why will that person not be able to do it?

Exercise $\PageIndex{33}$

Suppose someone wants to find two consecutive even integers that add to 139. Why will that person not be able to do it?

Answer: …because the sum of any even number (in this case, 2) o even integers (consecutive or not) is even and, therefore, cannot be odd (in this case, 139)

Exercise $\PageIndex{34}$

Three numbers add to 35. The second number is five less than twice the smallest. The third number is exactly twice the smallest. Find the numbers.

Exercise $\PageIndex{35}$

Three numbers add to 37. The second number is one less than eight times the smallest. The third number is two less than eleven times the smallest. Find the numbers.

Answer: 2, 15, 20

Exercises for Review

Exercise $\PageIndex{36}$

Find the decimal representation of $0.34992 \div 4.32$

Exercise $\PageIndex{37}$

A 5-foot woman casts a 9-foot shadow at a particular time of the day. How tall is a person that casts a 10.8-foot shadow at the same time of the day?

Answer: 6 feet tall

Exercise $\PageIndex{38}$

Use the method of rounding to estimate the sum: $4 \dfrac{5}{12} + 15 \dfrac{1}{25}$

Exercise $\PageIndex{39}$

Convert 463 mg to cg.

Answer: 46.3 cg

Exercise $\PageIndex{40}$

Twice a number is added to 5. The result is 2 less than three times the number. What is the number?

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