2.3 Use a Problem Solving Strategy

Izabela Mazur; Kim Moshenko

2. Solving Linear Equations and Inequalities

2.3 Use a Problem Solving Strategy

Learning Objectives

By the end of this section it is expected that you will be able to:

Translate to an equation and solve
Translate and solve applications

Translate to an Equation and Solve

To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the equals sign. In the next few examples, we will translate sentences into equations and then solve the equations.

EXAMPLE 1

Translate and solve: Eleven more than x is equal to 54.

Solution

Translate.
Subtract 11 from both sides.
Simplify.
Check: Is 54 eleven more than 43? $\begin{array}{ccc}\hfill 43+11& \stackrel{?}{=}\hfill & 54\hfill \\ \hfill 54& =\hfill & 54✓\hfill \end{array}$

TRY IT 1

Translate and solve: Ten more than x is equal to 41.

Show answer

$x+10=41; x=31$

EXAMPLE 2

Translate and solve: The number 143 is the product of $-11$ and y.

Solution

Begin by translating the sentence into an equation.

Translate.
Divide by $-11$ .
Simplify.
Check:	$\begin{array}{cccc}& \hfill 143& =& -11y\hfill \\ & \hfill 143& \stackrel{?}{=}& -11\left(-13\right)\hfill \\ & \hfill 143& =& 143✓\hfill \end{array}$

TRY IT 2

Translate and solve: The number 132 is the product of −12 and y.

Show answer

$132=-12y;y=-11$

EXAMPLE 3

Translate and solve: The quotient of $y$ and $-4$ is $68$ .

Solution

Begin by translating the sentence into an equation.

Translate.

Multiply both sides by $-4$ .

Simplify.

Check:

Is the quotient of $y$ and $-4$ equal to $68$ ?

Let $y=-272$ .

Is the quotient of $-272$ and $-4$ equal to $68$ ?

Translate.

$\frac{-272}{-4}\stackrel{?}{=}68$

Simplify.

$\phantom{\rule{1.3em}{0ex}}68=68✓$

TRY IT 3

Translate and solve: The quotient of $q$ and $-8$ is 72.

Show answer

$\frac{q}{-8}=72;q=-576$

EXAMPLE 4

Translate and solve: Three-fourths of $p$ is 18.

Solution

Begin by translating the sentence into an equation. Remember, “of” translates into multiplication.

Translate.
Multiply both sides by $\frac{4}{3}.$
Simplify.
Check:	Is three-fourths of p equal to 18?
Let $p=24.$	Is three-fourths of 24 equal to 18?
Translate.	$\frac{3}{4}\cdot\phantom{\rule{0.2em}{0ex}}24\phantom{\rule{0.2em}{0ex}}\stackrel{?}{=}18$
Simplify.	$\phantom{\rule{1.6em}{0ex}}18=18✓$

TRY IT 4

Translate and solve: Two-fifths of $f$ is 16.

Show answer

$\frac{2}{5}f=16;f=40$

Translate and Solve Applications

Most of the time a question that requires an algebraic solution comes out of a real life situation. To begin, that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.

We will start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for. For example, you might use q for the number of quarters if you were solving a problem about coins.

EXAMPLE 5

How to Translate and Solve Applications

The Alec family recycled newspapers for two months. The two months of newspapers weighed a total of 57 pounds. The second month, the newspapers weighed 28 pounds. How much did the newspapers weigh the first month?

Solution

In the second row, the first cell says “Step 2. Identify what we are asked to find.” The second cell says “What are we asked to find?” The third cell says: “How much did the newspapers weigh the 2nd month?” In the third row, the first cell says “Step 3. Name what we are looking for. Choose a variable to represent that quantity.” The second cell says “Choose a variable.” The third cell says “Let w equal weight of the newspapers the 1st month.” In the fifth row, the first cell says “Step 5. Solve the equation using good algebra techniques.” The second cell says “Solve.” The third cell contains the equation with 28 being subtracted from both sides: w plus 28 minus 28 equals 57 minus 28, with minus 28 written in red. Below this is w equals 29. In the sixth row, the first cell says “Step 6. Check the answer and make sure it makes sense.” The second cell says “Does 1st month’s weight plus 2nd month’s weight equal 57 pounds?” The third cell contains the equation 29 plus 28 might equal 57. Below this is 57 equals 57 with a check mark next to it. In the seventh and final row, the first cell says ‘Step 7. Answer the question with a complete sentence.” The second cell says “Write a sentence to answer ‘How much did the newspapers weigh the 2nd month?’” The third cell contains the sentence “The 2nd month the newspapers weighed 29 pounds.”

TRY IT 5

Translate into an algebraic equation and solve:

The Snider family has two cats, Zeus and Athena. Together, they weigh 23 pounds. Zeus weighs 16 pounds. How much does Athena weigh?

Show answer

7 pounds

HOW TO: Solve an application

Read the problem. Make sure all the words and ideas are understood.
Identify what we are looking for.
Name what we are looking for. Choose a variable to represent that quantity.
Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.

EXAMPLE 6

Abdullah paid $28,675 for his new car. This was $875 less than the sticker price. What was the sticker price of the car?

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.	“What was the sticker price of the car?”
Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let $s=$ the sticker price of the car.
Step 4. Translate into an equation. Restate the problem in one sentence.	$28,675 is $875 less than the sticker price
Step 5. Solve the equation.	$28,675 is $875 less than s $\begin{array}{c}\hfill \begin{array}{ccc}\hfill 28,675& \stackrel{}{=}\hfill & s-875\hfill \\ \hfill 28,675 + 875& =\hfill & s - 875 + 875 \hfill \\ \hfill 29,550 & = \hfill & s \end{array}\hfill \end{array}$
Step 6. Check the answer.	Is $875 less than $29,550 equal to $28,675? $\begin{array}{c}\hfill \begin{array}{ccc}\hfill 29,550-875& \stackrel{?}{=}\hfill & 28,675\hfill \\ \hfill 28,675& =\hfill & 28,675✓\hfill \end{array}\hfill \end{array}$
Step 7. Answer the question with a complete sentence.	The sticker price of the car was $29,550.

TRY IT 6

Translate into an algebraic equation and solve:

Jaffrey paid $19,875 for her new car. This was $1,025 less than the sticker price. What was the sticker price of the car?

Show answer

$20,900

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems”.

EXAMPLE 7

The sum of twice a number and seven is 15. Find the number.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.	the number
Step 3. Name. Choose a variable to represent the number.	Let $n=$ the number.
Step 4. Translate.
Restate the problem as one sentence.
Translate into an equation.
Step 5. Solve the equation.
Subtract 7 from each side and simplify.
Divide each side by 2 and simplify.
Step 6. Check.
Is the sum of twice 4 and 7 equal to 15?	$\begin{array}{ccc}\hfill 2\cdot 4+7& \stackrel{?}{=}\hfill & 15\hfill \\ \hfill 15& =\hfill & 15✓\hfill \end{array}$
Step 7. Answer the question.	The number is 4.

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

TRY IT 7

The sum of four times a number and two is 14. Find the number.

Show answer

3

Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

EXAMPLE 8

One number is five more than another. The sum of the numbers is 21. Find the numbers.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.	We are looking for two numbers.
Step 3. Name. We have two numbers to name and need a name for each.
Choose a variable to represent the first number.	Let $n={1}^{\mathrm{st}}$ number.
What do we know about the second number?	One number is five more than another.
	$n+5={2}^{\mathrm{nd}}$ number
Step 4. Translate. Restate the problem as one sentence with all the important information.	The sum of the 1^st number and the 2^nd number is 21.
Translate into an equation.
Substitute the variable expressions.
Step 5. Solve the equation.
Combine like terms.
Subtract 5 from both sides and simplify.
Divide by 2 and simplify.
Find the second number, too.


Step 6. Check.
Do these numbers check in the problem?
Is one number 5 more than the other?	$\phantom{\rule{1.6em}{0ex}}13\stackrel{?}{=}8+5$
Is thirteen 5 more than 8? Yes.	$\phantom{\rule{1.6em}{0ex}}13=13✓$
Is the sum of the two numbers 21?	$8+13\stackrel{?}{=}21$
	$\phantom{\rule{1.6em}{0ex}}21=21✓$
Step 7. Answer the question.	The numbers are 8 and 13.

TRY IT 8

One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

Show answer

9, 15

Now, we will use the problem solving strategy to solve some geometry problems.

EXAMPLE 9

The length of a rectangle is $32$ metres and the width is $20$ metres. Find a) the perimeter, and b) the area.

Solution

a)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the perimeter of a rectangle
Step 3. Name. Choose a variable to represent it.	Let P = the perimeter
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The perimeter of the rectangle is 104 metres.

b)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of a rectangle
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area of the rectangle is 60 square metres.

TRY IT 9

The length of a rectangle is $120$ yards and the width is $50$ yards. Find a) the perimeter and b) the area.

Show answer

a) 340 yd

b) 6000 sq. yd

EXAMPLE 10

Find the length of a rectangle with perimeter $50$ inches and width $10$ inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length of the rectangle
Step 3. Name. Choose a variable to represent it.	Let L = the length
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length is 15 inches.

TRY IT 10

Find the length of a rectangle with a perimeter of $80$ inches and width of $25$ inches.

Show answer

15 in.

EXAMPLE 11

The area of a rectangular room is $168$ square feet. The length is $14$ feet. What is the width?

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the width of a rectangular room
Step 3. Name. Choose a variable to represent it.	Let W = width
Step 4.Translate. Write the appropriate formula and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The width of the room is 12 feet.

TRY IT 11

The area of a rectangle is $598$ square feet. The length is $23$ feet. What is the width?

Show answer

26 ft

EXAMPLE 12

The perimeter of a rectangular swimming pool is $150$ feet. The length is $15$ feet more than the width. Find the length and width.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length and width of the pool
Step 3. Name. Choose a variable to represent it. The length is 15 feet more than the width.	Let $W=\text{width}$ $W+15=\text{length}$
Step 4.Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length of the pool is 45 feet and the width is 30 feet.

TRY IT 12

The perimeter of a rectangular swimming pool is $200$ feet. The length is $40$ feet more than the width. Find the length and width.

Show answer

30 ft, 70 ft

The formula for the area of a triangle is $A=\frac{1}{2}bh$ , where $b$ is the base and $h$ is the height.

To find the area of the triangle, you need to know its base and height.

EXAMPLE 13

Find the area of a triangle whose base is $11$ inches and whose height is $8$ inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the triangle
Step 3. Name. Choose a variable to represent it.	let A = area of the triangle
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area is 44 square inches.

TRY IT 13

Find the area of a triangle with base $13$ inches and height $2$ inches.

Show answer

13 sq. in.

EXAMPLE 14

The perimeter of a triangular garden is $24$ feet. The lengths of two sides are $4$ feet and $9$ feet. How long is the third side?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	length of the third side of a triangle
Step 3. Name. Choose a variable to represent it.	Let c = the third side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The third side is 11 feet long.

TRY IT 14

The perimeter of a triangular garden is $24$ feet. The lengths of two sides are $18$ feet and $22$ feet. How long is the third side?

Show answer

8 ft

EXAMPLE 15

The area of a triangular church window is $90$ square metres. The base of the window is $15$ metres. What is the window’s height?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	height of a triangle
Step 3. Name. Choose a variable to represent it.	Let h = the height
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The height of the triangle is 12 metres.

TRY IT 15

The area of a triangular painting is $126$ square inches. The base is $18$ inches. What is the height?

Show answer

14 in.

Key Concepts

To translate a sentence to an equation
1. Locate the “equals” word(s). Translate to an equal sign (=).
2. Translate the words to the left of the “equals” word(s) into an algebraic expression.
3. Translate the words to the right of the “equals” word(s) into an algebraic expression.
To solve an application
1. Read the problem. Make sure all the words and ideas are understood.
2. Identify what we are looking for.
3. Name what we are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

2.3 Exercise Set

In the following exercises, translate to an equation and then solve it.

Nine more than $x$ is equal to 52.
Ten less than m is $-14$ .
The sum of y and $-30$ is 40.
The difference of $n$ and $\frac{1}{6}$ is $\frac{1}{2}$ .
The sum of $-4n$ and $5n$ is $-82$ .
133 is the product of $-19$ and n.
The quotient of $b$ and $-6$ is 18.
Three-tenths of x is 15.
The sum of two-fifths and f is one-half.
The difference of q and one-eighth is three-fourths

In the following exercises, translate into an equation and solve.

Avril rode her bike a total of 18 miles, from home to the library and then to the beach. The distance from Avril’s house to the library is 7 miles. What is the distance from the library to the beach?
Eva’s daughter is 15 years younger than her son. Eva’s son is 22 years old. How old is her daughter?
For a family birthday dinner, Celeste bought a turkey that weighed 5 pounds less than the one she bought for Thanksgiving. The birthday turkey weighed 16 pounds. How much did the Thanksgiving turkey weigh?
Arjun’s temperature was 0.7 degrees higher this morning than it had been last night. His temperature this morning was 101.2 degrees. What was his temperature last night?
Ron’s paycheck this week was $17.43 less than his paycheck last week. His paycheck this week was $103.76. How much was Ron’s paycheck last week?

In the following exercises, solve each number word problem

The sum of a number and eight is 12. Find the number.
The difference of twice a number and seven is 17. Find the number.
Three times the sum of a number and nine is 12. Find the number.
One number is six more than the other. Their sum is 42. Find the numbers.
The sum of two numbers is $-45.$ One number is nine more than the other. Find the numbers.
One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.
One number is one more than twice another. Their sum is $-5.$ Find the numbers.

In the following exercises, find the a) perimeter and b) area of each rectangle.

The length of a rectangle is $85$ feet and the width is $45$ feet.
A rectangular room is $15$ feet wide by $14$ feet long.

In the following exercises, solve.

Find the length of a rectangle with perimeter $124$ inches and width $38$ inches.
Find the width of a rectangle with perimeter $92$ metres and length $19$ metres.
The area of a rectangle is $414$ square metres. The length is $18$ metres. What is the width?
The length of a rectangle is $9$ inches more than the width. The perimeter is $46$ inches. Find the length and the width.
The perimeter of a rectangle is $58$ metres. The width of the rectangle is $5$ metres less than the length. Find the length and the width of the rectangle.
The width of the rectangle is $0.7$ metres less than the length. The perimeter of a rectangle is $52.6$ metres. Find the dimensions of the rectangle.
The perimeter of a rectangle is $150$ feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.
The length of a rectangle is $3$ metres less than twice the width. The perimeter is $36$ metres. Find the length and width.
The width of a rectangular window is $24$ inches. The area is $624$ square inches. What is the length?
The area of a rectangular roof is $2310$ square metres. The length is $42$ metres. What is the width?
The perimeter of a rectangular courtyard is $160$ feet. The length is $10$ feet more than the width. Find the length and the width.
The width of a rectangular window is $40$ inches less than the height. The perimeter of the doorway is $224$ inches. Find the length and the width.

In the following exercises, solve using the properties of triangles.

A triangular flag has base of $8$ foot and height of $1.5$ feet. What is its area?
What is the base of a triangle with an area of $207$ square inches and height of $18$ inches?
The perimeter of a triangular reflecting pool is $36$ yards. The lengths of two sides are $10$ yards and $15$ yards. How long is the third side?
The perimeter of a triangle is $39$ feet. One side of the triangle is $1$ foot longer than the second side. The third side is $2$ feet longer than the second side. Find the length of each side.