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Algebraic word problems have always
been a challenge for students. This
site is designed to help students
understand the language of addition,
subtraction, multiplication, and equal.
The exercises are designed to
reinforce the vocabulary of algebraic
word problems and also to translate
beginner algebraic word problems into
mathematical expressions. This site
will be a stepping stone for the more
advanced algebraic word problems in
two variables.
Let’s begin to unfold the mystery of word
problems
•
Math Vocabulary
*Finding “x”
*Understanding addition
*Understanding subtraction
*Understanding multiplication
*Understanding equal
•
Translating
expressions
word
problems
*Word Problems (one unknown)
*Word Problems (two or more unknowns)
•
Practice Exercises
*Finding “x’
*Addition
*Subtraction
*Multiplication
into
mathematical
*Equal
*Vocabulary Review Exercise
*Algebraic Word Problems (one unknown)
*Algebraic Word Problems (two or more unknowns)
The first step in translating word problems into mathematical
expressions is to identify what we are looking for.
In a mathematical word problem, we use the letter “x” to represent the
item we are looking for.
Example 1:
A number reduced by 10 is 7. What is the
number?
In this example, we are looking for a number;
Therefore we write:
Let x = a number
Example 2:
In two years Jim will be 18 years old.
What is Jim’s present age?
In this example, we are looking for Jim’s age;
Therefore we write:
Let x = Jim’s age
Practice!
1.
2.
3.
4.
5.
Let x = Greg’s present age
Let x = the number
Let x = number of dollars
Let x = Brenda’s age
Let x = the number
In mathematics, the symbol used to represent addition is “+.” There
are many words in the English language that imply addition. The
most common word is “plus.”
Example 1:
Two plus two
can be expressed as:
2+2
There are many other words in the English language that imply
addition; such as:
More
Add
Deposit
And
Together
Consecutive
Profit
Increase
Sum
In 5 years (In the future)
5 years from now
Raise
(In the future)
practice!
In mathematics, the symbol used to represent subtraction is “-”.
There are many words in the English language that imply subtraction.
The most common word is “Minus.”
Example 1:
three minus two
can be expressed as:
There are many other words in the English language that imply
subtraction; such as:
Less
Difference
Reduce
…years ago
Decrease
Take away Reduce
(In the past)
Withdraw
Minus
Practice!
1. A number, less seven, is 15.
Subtracted
2. Two years ago I was double my present age.
3. A number reduced by eight, is 27.
4. The difference between 13 and a number is 9.
5. If I withdraw $500 from my account I will have a balance of
$300.
6. Three years ago, I was 31 years old.
7. A number minus 12 gives 19.
8. Ten decreased by a number is 6.
9. Five years ago I was 21 years old.
10. A number take away 11 is 10.
In mathematics, the symbol normally used to represent multiplication
is “x”. Since we are using the symbol “x” to represent the item
we are looking for, we will use the symbol “•” to represent
multiplication.
There are many words in the English language that imply
multiplication. The most common word is “times.”
Example:
three times two
can be expressed as:
3•2
There are many other words in the English language that imply
multiplication; such as:
Twice
Times
Triple
Double
½ “of” a value
Product
Multiplied by…
Practice!
1. Twice a number, plus two, is twelve.
2. Seven added to ten times a number is 47.
3. The product of ten and a number is 30.
4. Half of a number, decreased by three, is 0.
5. Triple a number, less six, is 19
6. A number multiplied by 4, is 36.
7. The product of two consecutive numbers is 19.
8. Ten added to 2/3 of a number is 13.
9. Twice a number plus seven is 39.
10. Eight times a number, less 12, is 52.
In mathematics, the symbol used to represent equal is “=.”
There are many words in the English language that imply equal.
Common words representing “=”
• Result
• Total
• Equal
Other words representing “=”
• Is, was, will be
• Or
• Will have
Practice!
Example:
In three years Paul will be double his present age. What is
Paul’s present age?
1st step:
Identify the variable “x”
In a word problem, “x” will represent the unknown; what you are
looking for.
Let x = Paul’s present age
2nd step:
Identify the key words.
tip #1
Identify the word(s) that represent “=”
In three years Paul will be double his present age.
=
tip #2
Identify the key words on the left side of the “=”
In three years Paul will be double his present age.
=
+3
x
tip #3
Identify the key words on the right side of the “=”
In three years Paul will be double his present age.
=
2•
3rd step:
Write the algebraic expression.
x
begin the algebraic expression with the variable “x”
In three years Paul will be double his present age.
x+3
=
4th step:
Solve the algebraic equation.
x + 3 = 2x
x – 2x + 3 = 0
-1x + 3 = 0
-1x = -3
x=3
Paul present age is 3 years old.
Practice!
2x
1. Three more than a number is 10. What is the number?
3
+
x
= 10
Let x = the number
x + 3 = 10
x = 10 – 3
x=7
The number is 7
2. Twice a number less 7 is 13. What is the number?
2•
x
- 7 = 13
Let x = the number
2x – 7 = 13
2x = 13 + 7
2x = 20
x = 10
The number is 10
3. Four years from now, Matthew will be 15 years old. What is
Matthew’s present age?
+4
x
=
15
Let x = Matthew’s present age
x + 4 = 15
x = 15 – 4
x = 11
Matthew is 11 years old
4. The difference between 13 and a number is 9. What is the
number?
13
x
=9
Let x = the number
13 – x = 9
-x = 9 – 13
-x = -4
x=4
The number is 4
5. A number, reduced by 5, is17. What is the number?
x
5 = 17
Let x = the number
x – 5 = 17
x = 22
The number is 22
6. Seven years ago, I was 39 years old. What is my present age?
-7
x = 39
Let x = my present age
x – 7 = 39
x = 46
I am presently 46 years old
7. If I add 5 to a number or I multiply the number by 2, I get the
same result.
+ 5
x
=
•
x
2
Let x = the number
x + 5 = 2x
x – 2x = -5
-1x = -5
x=5
The number is 5
8. If I deposit $90 in my savings account, I will have a total of
$210.
+
90
x
=
210
Let x = money in my savings
x + 90 = 210
x = 210 – 90
x = 120
I have $120 in my savings account
9. The sum of 14 and a number is 56.
+
14
x
= 56
Let x = the number
14 + x = 56
x = 56 – 14
x = 42
The number is 42
10.Fourteen years from now, Brenda will be double her present
age.
+ 14
x
=
2•
x
Let x = Brenda’s age
x + 14 = 2x
x – 2x = -14
-1x = -14
x = 14
Brenda is 14 years old.
Example 1:
Sam has 10 more dollars than Marc. Together they have
$90. How much money do Sam and Marc each have?
1st step:
Make a list of the items you are looking for.
1. Sam’s money
2. Marc’s money
2nd step:
Identify item (from the list) that represents “x”
In the first sentence of the word problem, we learn that Sam has $10
than Marc. We do not have a description for how much money Marc
has. Therefore, Marc must represent “x”
Marc’s money: x
Sam’s money: x + 10
3rd step:
Write the algebraic expression.
The second sentence will help you determine how to
combine the two unknowns.
“Together they have $90”
+
=
90
Therefore, we must add the two unknowns.
Xmarc + (x + 10)sam = 90
4th step:
Solve the algebraic equation.
x + x + 10 = 90
2x + 10 = 90 – 10
2x = 90 – 10
2x = 80
x = 40
5th step:
Solve for both unknowns.
Since we determined that:
Marc’s money: x
Marc has $40 and
Sam’s money: x + 10
Sam has 40 + 10 = $50
Example 2:
The sum of three consecutive numbers is 24. What are
the three numbers?
1st step:
Make a list of the items you are looking for.
1. the first number
2. the second number
3. the third number
2nd step:
Identify item (from the list) that represents “x”
When we are dealing with a “consecutive” word problem, the first item
of the list will always represent “x”
The first number: x
An example of three consecutive numbers is 11, 12, 13 Notice that
12 is one more than 11, and
13 is 1 more than 12.
Remember from the vocabulary section that consecutive means
“+”.
Therefore,
The first number: x
The second number: x + 1
The third number: x + 2
3rd step:
Write the algebraic expression.
The first sentence will help you determine how to combine
the three unknowns.
The sum of three consecutive numbers is 24.
+
=
24
Therefore, we must add the three unknowns.
x + (x + 1) + (x + 2) = 24
4th step:
Solve the algebraic equation.
x + (x + 1) + (x + 2) = 24
x + x + 1 + x + 2 = 24
3x + 3 = 24
3x = 24 – 3
3x = 21
x=7
5th step:
Solve for all unknowns.
Since we determined that:
The first number: x
and
The second number: x + 1
+ 1 = 8; and
The first number is 7;
The second number is 7
The third number: x + 2
2=9
The third number is 7 +
Therefore, the three consecutive number that add to
give 24 are 7,8,9.
Example 3:
The length of a rectangle is 3cm more than its width.
The perimeter of the rectangle is 18cm. What is the
length and width of the rectangle?
1st step:
Make a list of the items you are looking for.
1. the length
2. the width
2nd step:
Identify item (from the list) that represents “x”
In the first sentence of the word problem, we learn that the length is
3cm more than the width. We do not have a description for the width.
Therefore, the width must represent “x”
The width: x
The length: x + 3
3rd step:
Write the algebraic expression.
The second sentence will help you determine how to
combine the two unknowns.
The perimeter of the rectangle is 18cm.
=
18
When dealing with a geometry word problem, it is a helpful to draw a
picture with the information provided.
x+3
x
x
x+3
In geometry, we learned that to find the perimeter of any shape, we
must add up all the sides. Therefore,
x + (x + 3) + x + (x + 3) = 18
4th step:
Solve the algebraic equation.
x + (x + 3) + x + (x + 3) = 18
x + x + 3 + x + x + 3 = 18
4x + 6 = 18
4x = 18 – 6
4x = 12
x=3
5th step:
Solve for all unknowns:
Since we determined that:
The width: x
The length: x + 3
The width is 3cm
The length is 3 + 3 = 6cm
Practice!
1.
Let the first number = x
Let the second number = x + 1
x + (x + 1) = 31
x + x + 1 = 31
2x + 1 = 31
2x = 31 – 1
2x = 30
x = 15
The first number is 15 and the second number is 16
2.
Let the number of quarters = x
Let the number of dimes = x + 6
x + (x + 6) = 18
x + x + 6 = 18
2x = 18 – 6
2x = 12
x=6
I have 6 quarters and 12 dimes
3.
Let Clara’s age = x
Let Matthew’s age = x + 11
x + (x + 11) = 45
x + x + 11 = 45
2x + 11 = 45
2x = 45 – 11
2x = 34
x = 17
Clara is 17 years old and Matthew is 28 years old
4.
Let the length = x
Let the width = x – 4
x + (x – 4) + x + ( x – 4) = 16
x + x – 4 + x + x – 4 = 16
4x – 8 = 16
4x = 16 + 8
4x = 24
x=6
The length of the rectangle is 6cm and the width of the rectangle is
2cm
5.
Let the first number = x
Let the second number = x + 1
Let the third number = x + 2
Let the fourth number = x + 3
x + (x + 1) + (x + 2) + (x + 3) = 42
x + x + 1 + x + 2 + x + 3 = 42
4x + 6 = 42
4x = 42 – 6
4x = 36
x=9
The first number is 9, the second number 10, the third number is 11,
and the fourth number is 12
6.
Let the width = x
Let the length = 2x
x + 2x + x + 2x = 72
6x = 72
x = 12
The width of the rectangle is 12cm and the length is 24cm
7.
Let the number of 2¢ stamps = x
Let the number of 51¢ stamps = 2x + 3
x + (2x + 3) = 12
x + 2x + 3 = 12
3x + 3 = 12
3x = 12 – 3
3x = 9
x=3
I have 3, 2¢ stamps and I have 9, 51¢stamps
8.
Let the first odd number = x
Let the second odd number = x + 2
x + (x + 2) = 136
x + x + 2 = 136
2x + 2 = 136
2x = 136 -2
2x = 134
x = 67
The first odd number is 67 and the second is 69
9.
Let the first even number = x
Let the second even number = x + 2
Let the third even number = x + 4
x + (x + 2) + (x + 4) = 66
x + x + 2 + x + 4 = 66
3x + 6 = 66
3x = 66 – 6
3x = 60
x = 20
The first even number is 20, the second is 22 and the third is 24.
10.
Let the number of dimes = x
Let the number of nickels = x + 7
0.10(x) + 0.05(x + 7) = 2.75
0.10x + 0.05x +0 .35 = 2.75
0.15x + 0.35 = 2.75
0.15x = 2.75 – 0.35
0.15x = 2.40
x = 16
I have 16 dimes and 23 nickels in my piggy bank
Which of the following represents the underlined section of each
sentence?
1. In two years, David will be 18 years old.
a.
b.
c.
d.
-2
+2
•2
=2
2. In two years, David will be 18 years old.
a. –
b. +
c. •
d. =
2. A number, reduced by 5 is 10.
a.
b.
c.
d.
-5
+5
•5
=5
3. A number, reduced by 5 is 10.
a.
b.
c.
d.
– 10
+ 10
• 10
= 10
4. Double a number increased by 10 is 14.
a.
b.
c.
d.
2-x
2+x
2•x
=2
5. Five years ago, I was 30 years old.
a.
b.
c.
d.
–
+
•
=
6. Five years ago, I was 30 years old.
a.
b.
c.
d.
-5
+5
•5
=5
7. The difference between 13 and a number is 10.
a.
b.
c.
d.
–
+
x
=
8. The difference between 13 and a number is 10.
a.
b.
c.
d.
– 10
+ 10
x 10
= 10
9. If I deposit $150 in my savings account, I will have $400 in my account.
a.
– 150
b.
c.
d.
+ 150
x 150
= 150
10. If I deposit $150 in my savings account, I will have $400 in my account.
a.
b.
c.
d.
–
+
x
=
Solutions