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Intro to Word Problems
(Developmental Algebra)
The most common difficulty people have with word problems is substituting
English terms for algebraic symbols and equations. Below are a few example of
converting from words to Algebraic form.
English
Algebraic Form
Five more than x
x+5
A number added to 3
x+3
A number increased by 7
x+7
5 less than a number
x-5
A number decreased by 6
x-6
The difference between a number and 12
x -12
The difference between 8 and a number
8-x
Twice a number
2 × x or 2x
Three times a number
3 × x or 3x
Quotient of x and 3
x ÷ 3 or
x
3
Quotient of 3 and a number
3 ÷ x or
3
x
Four is two more than a number
4=x+2
The product of 5 and a number is 15
5 × x = 15 or 5x = 15
One half a number is 10
½ × x = 10 or
Five times the difference of a number and 9
5(x -9)
The sum of two consecutive integers
x + (x + 1)
The sum of two even consecutive integers
x + (x + 2)
The sum of two odd consecutive integers
x + (x + 2)
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x
= 10
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Intro to Word Problems
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Solving "Parts Problems"
(Developmental Algebra)
Some word problems are often called "parts" problems because the pieces or
"parts" add to a given total. In the problem, we're told something about one (or
more) of the parts (look for key words like “is”, “was”, “are” to identify given
information). Call the totally unknown "part" x. To solve word problems, the
following strategy can be used:
Problem Solving Strategy
1. Find the total. What do the parts add up to?
2. Find the parts. Which part do we know something about? Which part is
totally unknown?
a. Let "x" represent the unknown "part”.
b. Then use "x" to write an expression for the part we're told something
about.
3. Write an equation and solve. The total equals the sum of its parts. Use
the expressions from Step 2 to write a sum:
1st Part + 2nd Part = Total
4. State the answer. Re-read the problem. What were you asked to find?
To answer the question, substitute the solution into the expressions
written in Step 2 and simplify.
Example:
A board measuring 97 feet is cut into three pieces so that the length of the
second piece is two feet more than the first piece, and the third piece is
three times the length of the first piece. How long is each piece?
1. Find the total: In this example, the total is the measure of the board
97 ft.
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Intro to Word Problems
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2. Find the parts:
Information about the second and third pieces is given, which means
that we don’t have any information about the first piece. The first piece
is the unknown.
First piece
x
Second piece
x+2
Third piece
3x
2. Write an equation using this information and solve:
1ST Part + 2nd Part + 3rd Part = Total
x
+ x + 2 + 3x
= 97
5x + 2 = 97
5x = 95
x = 19
3. State the answer:
First piece is
19 ft.
Second piece is
19 + 2 = 21 ft.
Third Piece is
3(19) = 57 ft.
Example:
The perimeter of a rectangular field is 196 feet. Find the length and width
of the field if the length measures 8 feet longer than twice the width.
1. In this example the perimeter is the total
196 ft.
2. Information about the length is given, which means that the width is
the unknown. The width and the length are your parts.
First Part.
Width
x
Second Part. Length 2x + 8
3. The formula to find the perimeter of a rectangle is:
P = 2l + 2 w
Perimeter = 2(First part) + 2 (Second Part)
196 = 2(2x + 8)
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+ 2(x)
Intro to Word Problems
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196 = 4x + 16 + 2x
196 = 6x + 16
180 = 6x
x = 30
4. The width is
The length is
30 ft.
2(30) + 8
68 ft.
Other Examples:

The difference of three times a number and four is equal to the difference
of twice the number and six.
3x – 4 = 2x – 6, x = -2

The sum of the interior angles of a triangle is 180o. The second angle of a
triangle is three times as large as the first. The third angle is 80o greater
than the first. Find the measure of the first angle.
180 = x + 3x + x + 80, x = 20
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Intro to Word Problems
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Solving Problems with Inequalities
Many applications have more than one solution. For example, a student can
earn one of several scores on a final exam to receive a passing grade. To solve
these problems, we write an inequality. The table below lists the inequality
symbols and the common key words and phrases that tell us to use an
inequality sign instead of an equal sign.
Inequality
<
>
≤
≥
Key Words and
Phrases
Example
Translation
y<8
n < −2
is less than
is smaller than


y is less than 8
the number "n" is
smaller than −2
is greater than
is more than


is larger than

p is greater than 0
the distance "d" is more
than 4 miles
the base "b" is larger
than 18 meters
is less than or
equal to
is at most

is no more than

is greater than or
equal to
is at least

is no less than



x is less than or equal
to 5
the height "h" is at most
12 feet
the cost "c" is no more
than $100
q is greater than or
equal to 9
the score "s" must be at
least 70
the average "a" can be
no less than 75
p>0
d>4
b > 18
x≤5
h ≤ 12
c ≤ 100
q≥9
s ≥ 85
a ≥ 75
Example:
Mario wants to hold his wife’s birthday party at a restaurant. The
restaurant charges $50 per person plus a cleanup fee of $80. How many
people can he invite if he can spend no more than $600?
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Intro to Word Problems
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1. Find the total
No more than 600
≤ 600
2. Find the parts. In this type of problem, your parts are a fixed
amount and a changing amount in terms of x.
Fixed amount
= 80
Changing amount
= 50(1) for 1 person
50(2) for 2 people
50(x) for x many people
1. Solve
Fixed part + Changing amount no more than
80 + 50x ≤ 600
x ≤ 10.4
Mario can invite no more than 10 people
Other Example:
A couple wants to hold their reception at their church hall. The church
charges $15 per person plus a clean-up fee of $100. If they plan to
spend no more than $1000, how many people they can invite?
15x + 100 ≤1000
x ≤ 60 people
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Intro to Word Problems