3-3-1 One step equations
Up to this point, variables have only been used in expressions. These expressions have been
simplified. Sometimes the number represented by the variable was known and a value for the
expression could be determined.
Now, with an equation, the value for a variable can sometimes be determined.
Think of an equation as a scale. If you add something to one side, you must add the same amount to
the other side to make it balance.
If the items on one side of the balance are split in half then the items on the other side also need to be
split to make it balance.
The object of the puzzle is to get the x or other variable alone on one side of the equation.
X+ 245 = -986
X+ 245 = -986
X+ 245 = -986
-245
-245 -245
X+ 0
X = -1231
Do the opposite.—if something is added, subtract
What is done to one side of the equation, do to the other. This keeps the balance.
The object of the puzzle is to get the variable by itself on one side of the equation.
2 1/3 = z – 4 1/5
2 1/3= z – 4 1/5
What you do
2 1/3= z – 4 1/5
+ 4 1/5
here, do here.
+ 4 1/5
+ 4 1/5
8
z+0
6 /15 = z
Do the opposite. –if something is subtracted, add.
What is done to one side of the equation, do to the other.
Fill in the blanks for the following. Follow the pattern. Write the number to be added or subtracted on
both sides of the equation. Draw the lines. Do not just see it in your mind. Do the arithmetic by hand
for the practice. The object of the puzzle is to get the variable by itself on one side of the equation.
Do the opposite.
What is done to one side of the equation, do to the other.
W+867= 234
-867 -867
W = -633
68
f – 23 = -41
f=
-342= 256+ y
=y
t-98 = -786
t=
m+ 2/5= 2 3/4
f –2 1/3= - 4 3/4
-3 1/2=6 2/3+y
f=
=y
m=
Practice: Solve the following one step equations.
a) 7+s=15
21 =506 + x
b)
a – 3/4 = 5/6
c)
3 4/5 = y – 2/5
d)
2 1/2 = -3 2/3 + b
b-9.8 = -0.87
b=
R – 8 = 56
34 = e - 97
7 2/3+r=5 ½
0.21 =5.6 + p
f – 8 4/9 = - 5 5/12
-0.21 = 0.023 + x
e – 0.768 = 5
/3 = w + 9 2/5
-3.4 = d – 9.7
0.675 + v=1.5
2
3 2/5+w= - 6 1/2
The object of the puzzle is to get the variable by itself on one side of the equation.
Do the opposite. –if multiplied, divide.
What is done to one side of the equation, do to the other.
The 5 is multiplied by the r so divide.
5r 510
5r 510
5r  510 

 
5
5
5
51
t =
4 1/2 = 3s
scratch work
1
8
3
1
3

5
8

10

3
5
8

3
10

2
3
e = 17
5
16
x3
16
5e
 5  17
5
s = 14
8
6
b)
c)
 18
0.05 y  15
4 2 3  7g
2814  7g
3 y  159
15  f
27 
3
21
s  14
5
Do the fraction division for the practice.
Then look at the example for an easier
way.
1
5
3 x
3
8
10
5
x
3
8
1
e=85
Practice:
a) q
 r  102
15t = 54
1
5
3 x
1
5
3 x  3  8
1
1
3
8
3
3
3
3
5
102
 3  10
5  3 
x



 10  3
8  10 2 


3
x
16
1.Change fractions to improper
2. Multiply both sides by the
reciprocal.
2
s 8
3
3
14  x
5
81
x
4
2 3
2  g
5 4
7
21
s
18
81
u
 1.8
6.8
3 y  15
y
 10
1
2
2
69
3-3-2 Plotting Solutions
A visual solution is sometimes more useful than only having the numerical answer written down. This
will become more apparent in future lessons.
3 x  12
3 x 12

3
3
x  4
The solution is plotted on a number line.
3 x  1
3 x 1

3
3
x   13
When an answer lies between integers, draw a line to indicate where the answer is and
write the answer below the dot.
 13
Practice: Solve the following one step equations and plot the solution. Do the arithmetic without a
calculator for the fraction practice.
a) y - 72/3= -3 2/5
b)
3 2 5  7 1 2 n
c)
7x
 1.4
5
d)
7 3 4  p  10 7 8
e)
2 13  1 3 4  y
f)
r  1.6  2 3 4
70
3-3-3 Simple Percentage Equations
Percent is per hundred. Imagine 15 red marbles in a pile of 40 marbles. If the ratio is the same, how
many red marbles would there be in a pile of 100 marbles?
In an equation
A percentage is a portion of the whole amount. Think of a 50% off
write = for “is” and
sale on a $24 shirt. To figure this divide by 2 or multiply by .5.
multiply for “of.”
Review topic 2 showed that to change a percentage to a decimal,
Keep the numbers the same.
divide by 100. This is easily done by moving the decimal to the left
Write the percent as a
two places. 50% is .5.
decimal.
Put an x for the “what”
50% of $24 can be written .5(24). The result is 12. Of means times
in math.
Review solving one step equations by rewriting a percent question sentence as an equation.
What
is
20%
of
15?
Solve this equation by multiplying .2 and 15.
x=.20(15)





x= 3
x
=
.20
15

14%
of
what
is
70?





.14

x
=
70
12
is
15%
of
what?





12
=
.15

x
2/3%
of
what
is
7 ½?





2/300

x
=
7½
What
is
33 1/3 %
of
1 ½?




x
=

33 13
100

1½
/5 %
of
what?



/5100

x
1 2/3
is


1 /3
=
2
2
2
.14 x  70
.14 x 70

.14 .14
x  500
12  .15 x
12 .15 x

.15 .15
80  x
Because there are
fractions in the
problem, change
the percent to a
fraction instead of
a decimal.
2
x  7 12
300
 300  2
 300 
x  7 12 



 2  300
 2 
x
15  300  15  3 0 0 75 
  1125

 
2  2  2 1  2 1 
Change 33 1/3 to a fraction.
33 13  100  1003  1001  100 3  1100 
x  13 1 1 2  13  3 2  1 2
1
3
1 2 3  1 250 x
1 2 3 1250 x

1
1
250
1 3
2
1
250
250
 5 3  2501  83 13  x
71
Practice: Write the percentage question as a one step equation, then solve. Divide each percent by
100 before writing the equation.
3
a)
34%
of
what
is
16?
/4%
of
what
is
65?

b)
c)









21
is
15%
of
what?
2.1
is
.5%
of
what?










What
is
15%
of
32?





Do the long division to review decimal division.
What
is
21/5%
of
3.2?





Either change the fraction to a decimal or the
decimal to a fraction.
More examples:
12
is
what percent
of
90?





12
=
x

90
12  x(90)
12 90 x

Cancel.
90 90
2
x
15
When a percent is put into
an equation, it is changed
into a fraction or a decimal by
dividing by 100.
To get a percent out of an
equation, multiply by 100.
2
What percent
of
30
is





x

x
=
10
is
what percent
of
16?





e)
100  215  1001  40 3 %
30 x  10
10?
Practice:
d)
3
15
30 x 10 1

 3
30 30
1  100  100  33 1 %
3
3
3
/4
is
what percent
of
16?





3
What percent
of
40
is
600?
What percent
of
3
/4
is
6?










Remember to multiply the result from the equation by 100 and add the percent symbol.
f) What is 20% of 32?
g) What percent of 15 is 300?
h) 20% of what is 36?
i) 54 is what percent of 100?
72
14 is what percent of 84?
32 is 55% of what?
350% of 22 is what?
17% of what is 22?
What percent of 2/3 is 16?
10% of 15 is what?
3 ¼ is 25% of what?
What is 2 ½ % of 25?
Reword word problems to mimic the easier
a) Three hundred entered Saturday’s
jog-a-thon. 5 didn’t finish. What
percent didn’t finish?
Reword: What percent of 300 is
5?
questions already done. Then solve the problem.
24% of the runners wrapped their bad knee. Out
of the 575 runners, how many had a wrapped
knee?
96% of the runners finished last year.
575 runners finished. How many
entered?
b) A local library has 150,000 books.
An average of 7% are checked
out. About how many are
checked out?
Little Johnny has 50 books in his little library. He
loaned Sally 12 books. What percent did Sally
borrow?
If 2.9% of the books in another library
are checked out and records show
12,350 are currently out, how many
books does the library have total?
c)
Out of a typical month Jonathan spends 85% of
his time with customers. When he works 125
hours in a month, How much time was spent with
customers?
Tony worked 14 hours one week with
customers. The rest of his time was
spent filling orders. If he spends 28%
of his time with customers, how many
hours did he work that week?
d) 2.5% of the what-cha-ma –call-its
produced an Saturday were
defective. There were 325
defective what-cha-ma –call-its.
How many useable what-cha-ma
–call-its were produced?
A week later the same company made 5,090
what-cha-ma –call-its. 35 were defective. What
percent were bad? Which week was better?
Across town a rival company has a
0.3% defective rate. They produce
3402 per day. How many are
defective?
e)
Jill missed 4 and got a 72% on a
test. How many were on the
test?
Jack earned an 85% on a test with 30 questions
How many did he get wrong?
Janet missed 3 on a test with 45
questions. What is her percent
correct?
f)
Tammy spent $52 on a dress that
had been marked down 35%.
What was the original price of the
dress?
A matching scarf is $12. If Tammy waits until it
goes on sale next week it will be 20% off. How
much will she save?
Tammy’s husband wants a new tool set
originally priced at $45, on sale for
$30. What percent off is the tool set?
He says it will go well with the dress
too.
I spend 5.5 out of 8 hours
working with others and 2.5
hours independently. What
percent do I spend on my own?
73
3-3-4 Word Problems
Dissect each question. "Is" usually means equals, and "of" is times.
Other words to know are sum - the answer to an addition problem, difference - the answer to a
subtraction problem, product - the answer to a multiplication problem, and quotient - the answer to a
division problem.
Example: 3/8 of a number is 15. What is the number? "3/8 of" says "3/8 times."
(3/8)n=15
Multiply both sides by 8/3. n=40
Practice: Each of the following can be solved as a one-step equation.
a) 45 is 62 more than a number. What is
3 times a number is 54. Find the number.
the number?
b) What is a number if 15 times the
number is 250?
4 more than a number is 12.
What is the number?
c) 12 and a number is 534.
What is the number?
126 is a number times 9.
What is the number?
d) 2/3 of a number is 108.
What is the number?
What is a number if the number
and 423 is negative 34?
e) -5 and a number is -15.
What is the number?
f) 3 2/5 and a number is 1 3/8.
What is the number?
g) The sum of a number and 2/5 is 3/8.
What is the number?
h) The difference of a number and 15 is 27.
What is the number?
-35 is a number times -5.
What is the number?
- 5 ¾ is a number added to 3 3/8.
What is the number?
15 is the sum of a number and 3 ½.
What is the number?
The difference of 45 and a number is 27.
What is the number?
(Technically this is a two step equation.)
i) What is a number that when added to
105 sums to 24?
j) 4/5 of a number is 7/12.
What is the number?
k) 42 and a number is -75.
What is the number?
74
A number is subtracted from 45.
The result is 23?
4 2/3 times a number is - 8 ¾.
What is the number?
What number when added to negative five
hundred results in 34?
75