We begin solving word problems with practice translating between an English sentence and an algebraic equation. First, spend a minute to recall some of the key words that repre- sent addition, subtraction, multiplication, and division. See Table 3-4.
Table 3-4
Addition: a + b Subtraction: a − b The sum of a and b
a plus b b added to a b more than a a increased by b The total of a and b
The difference of a and b a minus b
b subtracted from a a decreased by b b less than a Multiplication: a ã b Division: a ữ b The product of a and b
a times b a multiplied by b
The quotient of a and b a divided by b b divided into a The ratio of a and b a over b
a per b Read the
problem completely.
Assign labels to unknown quantities.
Write an equation in words.
Write a mathematical
equation.
Solve the equation.
Interpret the results and write the final
answer in words.
Translating a Sentence to a Mathematical Equation
A number decreased by 7 is 12. Find the number.
Solution:
Let x represent the number.
A number decreased by 7 is 12.
x − 7 = 12
x − 7 = 12 x − 7 + 7 = 12 + 7
x = 19 The number is 19.
Skill Practice
1. A number minus 6 is −22. Find the number.
Example 1
Step 1: Read the problem completely.
Step 2: Label the unknown.
Step 3: Write the equation in words Step 4: Translate to a mathematical
equation.
Step 5: Solve the equation.
Add 7 to both sides.
Step 6: Interpret the answer in words.
Avoiding Mistakes
To check the answer to Example 1, we see that 19 decreased by 7 is 12.
Translating a Sentence to a Mathematical Equation
Seven less than 3 times a number results in 11. Find the number.
Solution:
Example 2
Step 1: Read the problem completely.
Step 2: Label the unknown.
Step 3: Write the equation in words.
Step 4: Translate to a mathematical equation.
Step 5: Solve the equation. Add 7 to both sides.
Simplify.
Divide both sides by 3.
3x = 18 3x ___3 = 18 ___3 x = 6
The number is 6. Step 6: Interpret the answer in words.
Skill Practice
2. 4 subtracted from 8 times a number is 36. Find the number.
Let x represent the number.
Seven less than 3 times a number results in 11.
3x − 7 = 11
3x − 7 + 7 = 11 + 7
three times a number
7 less than results in 11
Avoiding Mistakes
To check the answer to Example 2, we have:
The value 3 times 6 is 18, and 7 less than 18 is 11.
Answers 1. The number is −16.
2. The number is 5.
Translating a Sentence to a Mathematical Equation
Two times the sum of a number and 8 equals 38. Find the number.
Solution:
Let x represent the number.
Two times the sum of a number and 8 equals 38.
2 ⋅ (x + 8) = 38
2(x + 8)
=
38
2x + 16
=
38 2x + 16 − 16
=
38 − 16 2x = 22 2x _____________________________
2
=
22 _____________________________
2
x
=
11
The number is 11.
Skill Practice
3. 8 added to twice a number is 22. Find the number.
Example 3
Step 1: Read the problem completely.
Step 2: Label the unknown.
Step 3: Write the equation in words.
Step 4: Translate to a mathematical equation.
Step 5: Solve the equation.
Clear parentheses.
Subtract 16 from both sides.
Simplify.
Divide both sides by 2.
Step 6: Interpret the answer in words.
the sum of a number and 8 two times
equals 38
⏟
Avoiding Mistakes
The sum (x + 8) must be enclosed in parentheses so that the entire sum is multiplied by 2.
Avoiding Mistakes
To check the answer to Example 3, we have:
The sum of 11 and 8 is 19. Twice this amount gives 38 as expected.
3. Applications of Linear Equations
In Example 4, we practice representing quantities within a word problem by using variables.
Representing Quantities Algebraically
a. Kathleen works twice as many hours in one week as Kevin. If Kevin works for h hours, write an expression representing the number of hours that Kathleen works.
b. At a carnival, rides cost $3 each. If Alicia takes n rides during the day, write an expression for the total cost.
c. Josie made $430 less during one week than her friend Annie made. If Annie made D dollars, write an expression for the amount that Josie made.
Solution:
a. Let h represent the number of hours that Kevin works.
Kathleen works twice as many hours as Kevin.
2h is the number of hours that Kathleen works.
⏞ Example 4
Answer 3. The number is 7.
©Purestock/PunchStock
In Examples 5 and 6, we practice solving application problems using linear equations.
Avoiding Mistakes
In Example 5, the two pieces should total 10 ft. We have, 2 ft + 8 ft = 10 ft as desired.
Applying a Linear Equation to Carpentry
A carpenter must cut a 10-ft board into two pieces to build a brace for a picnic table. If one piece needs to be four times longer than the other piece, how long should each piece be?
Solution:
Example 5
Step 1: Read the problem completely.
Step 2: Label the unknowns. Draw a picture.
10 ft
4x x
Step 3: Write an equation in words.
Step 4: Write a mathematical equation.
Step 5: Solve the equation.
Combine like terms.
Divide both sides by 5.
Step 6: Interpret the results in words.
Recall that x represents the length of the shorter piece. Therefore, the shorter piece is 2 ft. The longer piece is given by 4x or 4(2 ft) = 8 ft.
The pieces are 2 ft and 8 ft.
Skill Practice
7. A piece of cable 92 ft long is to be cut into two pieces. One piece must be three times longer than the other. How long should each piece be?
We can let x represent the length of either piece. However, if we choose x to be the length of the shorter piece, then the longer piece has to be 4x (4 times as long).
Let x = the length of the shorter piece.
Then 4x = the length of the longer piece.
( Length ofone piece ) +
( length of theother piece ) =
( total length )
x
+
4x
=
10
x + 4x
=
10 5x
= 10 5x _____________________________
5
= 10 _____________________________
5 x
=
2
Answers
4. 3x 5. 7d 6. x + 8 7. One piece should be 23 ft and the
b. Let n represent the number of rides Alicia takes during the day.
Rides cost $3 each.
3n is the total cost.
c. Let D represent the amount of money that Annie made during the week.
Josie made $430 less than Annie.
D − 430 represents the amount that Josie made.
Skill Practice
4. Tasha ate three times as many M&Ms as her friend Kate. If Kate ate x M&Ms, write an expression for the number that Tasha ate.
5. Casey bought seven books from a sale rack. If the books cost d dollars each, write an expression for the total cost.
6. One week Kim worked 8 hr more than her friend Tom. If Tom worked x hours, write an expression for the number of hours that Kim worked.
⏞
⏞
Answer
8. The Craftsman model costs $190 and the DeWalt model costs $500.
TIP: In Example 6, we could have let x represent either the cost of the 32″ TV or the 42″ TV.
Suppose we had let x represent the cost of the 42″ model.
Then x − 800 is the cost of the 32″ model (the 32″ model is less expensive).
( Cost of 32″ ) + ( cost of 42″ )
=
( total cost )
x − 800 + x
=
2000 2x − 800 = 2000 2x − 800 + 800
=
2000 + 800 2x
=
2800
x
=
1400
Therefore, the 42″ TV costs $1400 as expected.
The 32″ TV costs x − 800 or $1400 − $800 = $600 .
Applying a Linear Equation
One model of a 42″ HDTV sells for $800 more than a smaller 32″ HDTV. The combined cost for these two televisions is $2000. Find the cost for each television.
Solution:
Let x represent the cost of the 32″ TV.
Then x + 800 represents the cost of the 42″ TV.
( Cost of 32″ ) + ( cost of 42″ )
=
( total cost )
x + x + 800
=
2000
x + x + 800
=
2000 2x + 800 = 2000 2x + 800 − 800
=
2000 − 800 2x
=
1200
2x _____________________________
2
=
1200 _____________________________
2
x
=
600
Since x = 600, the 32″ TV costs $600.
The cost of the 42″ model is represented by x + 800 = $600 + $800 = $1400 . Skill Practice
8. A kit of cordless 18-volt tools made by Craftsman cost $310 less than a similar kit made by DeWalt. The combined cost for both models is $690. Find the cost for each model. (Source: Consumer Reports)
Example 6
Step 1: Read the problem completely.
Step 2: Label the unknowns.
Step 3: Write an equation in words.
Step 4: Write a mathematical equation.
Step 5: Solve the equation.
Combine like terms.
Subtract 800 from both sides.
Divide both sides by 2.
Step 6: Interpret the results in words.
©Alamy
Study Skills Exercise
In solving an application it is very important first to read and understand what is being asked in the problem. One way to do this is to read the problem several times. Another is to read it out loud so you can hear yourself. Another is to rewrite the problem in your own words. Which of these methods do you think will help you in understanding an application?
Review Exercises
1. Use substitution to determine if 3 is a solution to the equation 4x + 1 = 11 . 2. Use substitution to determine if −4 is a solution to the equation −3x + 9 = 21 . For Exercises 3–8, solve the equation.
3. 3t − 15 = −24 4. −6x + 4 = 16 5. __ b
5 − 5 = −14 6. __ w
8 − 3 = 3 7. 2x + 22 = 6x − 2 8. −5y − 34 = −3y + 12