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Lesson 34 ! page 1
Lesson 34
Algebra with Decimals
Expressions and equations with decimal numbers occur in financial and scientific situations. Isn’t it great that we know all the
rules we previously learned will stay the same?
Simplifying Expressions with Decimals
We can combine like terms and use the distributive property to simplify expressions with decimal numbers, just as we did
with whole numbers, signed numbers, and fractions.
Example: Simplify each expression.
30x + 7x = 37x
(
3 2x + 5
3x + 0.7x = 3.7x
)
(
0.3 2x + 5
= 3(2x ) + 3(5) = 6x + 15
(
)
0.3x + 0.07x = 0.37x
)
(
0.3 0.2x + 0.5
= 0.3(2x ) + 0.3(5)
= 0.6x + 1.5
(
)
= 0.3(0.2x ) + 0.3(0.5)
= 0.06x + 0.15
)
!2 5x ! 1 + 7x
!0.2 0.5x ! 1 + 0.07x
= !10x + 2 + 7x
= !3x + 2
= !0.1x + 0.2 + 0.07x
= !0.03x + 0.2
5 y ! 0.25x + 0.544 y + 7.05x
5.544 y + 6.8x
Solving Equations with Decimals
Some or all of the numbers in an equation may be written in decimal form. When constants in the equation are written as
decimals, the answer should also be written as a decimal.
Example: Solve the equations.
x + 0.3 = 5
x + 0.3 = 5
x + 0.3 ! 0.3 = 5 ! 0.3
x = 4.7
0.2x = 16
0.2x = 16
x = 80
0.2x / 0.2 = 16 / 0.2
0.2x + 0.5 = 0.6
0.2x + 0.5 = 0.6
0.2x = 0.1
x = 0.5
© 2010 Cheryl Wilcox
x ! 4 = 1.55
x ! 4 = 1.55
x = 5.55
x ! 4 + 4 = 1.55 + 4
x
= 9.68
2
x
• 2 = 9.68 • 2
2
x = 19.36
0.75 ! x = 2.08
0.2x + 0.5 ! 0.5 = 0.6 ! 0.5
0.2x / 0.2 = 0.1/ 0.2
!1x + 0.75 = 2.08
!1x + 0.75 ! 0.75 = 2.08 ! 0.75
!1x = 1.33
x = !1.33
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Lesson 34 ! page 2
Dropping the Decimal Point
When all the constants in an equation are written as fractions with the same denominator, the denominator can be dropped.
(See Lesson 30). If all the constants in an equation are written with the same number of decimal places, their fraction forms
will have the same denominator, which can be dropped. This is equivalent to dropping the decimal point from the equation
and having only whole numbers. The answer may still be a decimal, though.
For example, we can solve the equation with fractions either straightforwardly or by dropping the denominator:
3
2
5
x!
=
10
10 10
3
2
5
x!
=
10
10 10
Straightforward approach:
Every constant in the equation has denominator 10, so
dropping the denominator is equivalent to multiplying both
sides of the equation by 10.
3
2
5
x!
=
10
10 10
3
7
x=
10
10
7
x=
3
3
2
2
5
2
x!
+
=
+
10
10 10 10 10
10 3
7 10
•
x=
•
3 10
10 3
3x ! 2 = 5
3x ! 2 + 2 = 5 + 2
3x = 7
x=
3x / 3 = 7 / 3
7
3
The same equation written with decimals can also be solved either straightforwardly or by dropping the decimal point:
0.3x ! 0.2 = 0.5
0.3x ! 0.2 = 0.5
Straightforward approach:
Every constant in the equation has one decimal place, the
tenths place. Dropping the decimal point is equivalent to
multiplying both sides of the equation by 10.
0.3x ! 0.2 = 0.5
0.3x = 0.7
7
x=
3
0.3x ! 0.2 + 0.2 = 0.5 + 0.2
0.3x / 0.3 = 0.7 / 0.3
3x ! 2 = 5
3x = 7
x=
3x ! 2 + 2 = 5 + 2
3x / 3 = 7 / 3
7
3
There are some CAUTIONS to pay attention to if you decide to use the second method (called clearing decimals).
1. EQUATIONS ONLY! Dropping the denominator or the decimal point can only be done when solving equations,
never when simplifying expressions. The method depends on multiplying both sides of an equation by the same
number. When simplifying an expression, there are not two sides to keep in balance.
2. EVERY NUMBER! Every number in the equation must have the same denominator or the same number of
decimal places. Be especially careful not to overlook whole numbers in the equation. Everything must be converted
to the same denominator or the same number of decimal places.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 34 ! page 3
Example: Solve the equations by clearing decimals.
0.08x ! 0.1= 0.2
0.22 ! x = 0.04
This equation has 0.08, with two decimal places
(denominator 100) AND 0.1 and 0.2, with only one
decimal place (denominator 10). You can always put
zeros at the end of a decimal without changing its
value, so re-write the equation so that all the decimals
have two places:
Looks good, right? All the numbers we can see have two decimal
places. But what about the coefficient of x? If we re-write the
equation, we find a hidden, invisible –1:
0.08x ! 0.10 = 0.20
!1.00x + 0.22 = 0.04
Now you can drop the decimal points:
Dropping the decimal points:
8x ! 10 = 20
!100x + 22 = 4
8x ! 10 = 20
8x ! 10 + 10 = 20 + 10
8x = 30
8x / 8 = 30 / 8
x = 3.75
!100x + 22 = 4
! 100x + 22 ! 22 = 4 ! 22
!100x = !18
! 100x / !100 = !18 / !100
x = 0.18
!1x + 0.22 = 0.04
That –1 will need two decimal places as well:
If you practice the process a bit, you’ll eventually be able to use your own judgment about when it will save you work. My
opinion is that it saved work on the first example, but increased the work in the second.
Solving Application Problems with Decimals
Word problems often involve the use of decimals in equations.
Example: Use the formula to write an equation to solve the problem.
Profit = Revenue – Cost
Net Pay = Gross Pay – Deductions
P = R !C
N = G !D
Use the formula to find the costs of a business that had
revenue of $67,892.91 and made a profit of $18,766.34.
P = R !C
18,766.34 = 67,892.91! C
!1C + 67,892.91= 18,766.34
!67,892.91= !67,892.91
!1C = !49,126.57
C = 49,126.57
The costs were $49,126.57
Gross Pay is your full salary or full pay at your hourly rate.
Deductions include taxes, benefits, etc. Net Pay is the
amount you receive in your paycheck.
Angelica’s deductions were $345.18, and her net pay was
$1584.75. What was Angelica’s gross pay?
N = G !D
1584.75 = G ! 345.18
G ! 345.18 = 1584.75
+345.18 = +345.18
G = 1929.93
Angelica’s gross pay was $1929.93.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 34 ! page 4
Example: Use the formula to write an equation to solve the problem. Round answers if necessary.
Package Size / Number of Servings = Serving Size
Package Size / Number of Servings = Serving Size
P
=S
N
P
=S
N
Use the formula to find the serving size if a box of cereal
weighs 425 g and there are 14 servings per box.
S=
Use the formula to find the size of a box of cereal if there are
12 servings per box and each serving is 28.5 g.
425 g
! 30.4 g
14
28.5 =
P
12
P
= 28.5
12
P = 342
Each serving is about 30.4 g.
12 •
P
= 28.5 • 12
12
The box weighs 342 g.
Gas mileage = Miles Driven / Gallons Used
E=
Gas mileage = Miles Driven / Gallons Used
M
G
E=
Find the gas mileage if you drove 288.6 miles and used 12.8
gallons of gas.
E=
M
G
If your gas mileage is 18.4 mpg and you have 17.7 gallons
of gas, how many miles can you drive?
288.6 miles
! 22.5 mpg
12.8 gallons
18.4 mpg =
M
17.7 gals
M
M
= 18.4
17.7 •
= 18.4 • 17.7
17.7
17.7
M = 325.68
You can drive about 325 miles.
Note that it is more practical to round down this answer. You
don’t want to actually run out of gas.
The circumference of a circle is 154.2 cm, measured to the nearest tenth. What is the radius of the circle, to the nearest
tenth?
C = 2!r
154.2 = 2!r
2!r = 154.2
( )
( )
2!r / 2! = 154.2 / 2!
Be careful when dividing by 2!. If you don’t put it in
parentheses (2!), your calculator will not find the correct
result.
r " 24.5
The radius is about 24.5 cm.
!
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 34 ! page 5
Formulas So Far
Perimeter
inches, feet, miles;
centimeters, meters, kilometers
Geometric
Formulas
Area
ft2, mi2;
2
cm , m2, km2
Volume
in3, ft3, mi3;
cm3, m3, km3
in2,
Rectangle
Box
(
)
A = LW
P = 2 L + W = 2L + 2W
Triangle
A=
P = a +b +c
V = LWH
1
bh
bh =
2
2
( )
Circle
Sphere
C = !d = 2!r
A = !r 2
V=
(The perimeter of a circle
is called the circumference.)
Rates
Rate (Speed) is Distance over Time
r=
d
t
miles =
miles
• hours
hour
feet =
Other
rates:
items =
items
• boxes
box
gas mileage =
C=
(
5 F ! 32
9
) = (F ! 32) / 1.8
Celsius to Fahrenheit
9
F = C + 32 = 1.8C + 32
5
© 2010 Cheryl Wilcox
The related multiplication is
Distance = Rate • Time
d = rt
Examples
with
units:
Fahrenheit to Celsius
4 3
!r
3
feet
• seconds
second
miles
gallon
actions =
actions
• minutes
minute
servings per container =
amount per container
amount per serving
Profit = Revenue – Cost P = R ! C
Total Cost =
Net Pay = Gross Pay – Deductions
Cost per item • Number of Items
+ Fixed Costs
P = G !D
T = CN + F
Height of an object in feet t seconds
after falling (until it hits):
h = !16t 2 +
(
initial
velocity
)t + (
initial
height
)
Free Pre-Algebra
Lesson 33 ! page 6
Lesson 33: Algebra with Decimals
Worksheet
Name ___________________________________
1. Simplify by combining like terms.
2. Simplify using the distributive property.
0.5k ! 1.6 ! 0.02k + 2.8
0.03 1000 ! x
(
)
3. Simplify.
(
Solve the equations straightforwardly, without clearing decimals.
4. 1.1x ! 0.9 = !0.6
5. 0.4x ! 1= 0.6
Solve by first clearing decimals.
6. 1.1x ! 0.9 = !0.6
7. 0.4x ! 1= 0.6
8. Circle the method you prefer in problems 4 and 6. Circle the method you prefer in problems 5 and 7.
© 2010 Cheryl Wilcox
)
0.03 1000 ! x + 0.04x
Free Pre-Algebra
Lesson 33 ! page 7
Solve using a formula from page 5.
9. A rectangle has perimeter 9.88 cm. The length is 2.15 cm.
What is the width?
10. The circumference of a circle is approximately 45.8 cm.
Find the radius to the nearest tenth of a cm.
11. A truck traveled 280.8 miles at an average speed of
62.4 mph. How many hours did the trip take?
12. The label said that there were 18 servings per bag, and
that each serving was 1.6 ounces. How many ounces did the
bag hold?
13. The company’s costs were $74,911.18. Revenues were
$65,921.88. What profit did the company make?
14. The Celsius temperature was –11.2ºC. What was the
equivalent Fahrenheit temperature?
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 34 ! page 8
Lesson 34: Algebra with Decimals
Homework 34A
Name ________________________________________
Solve the equations.
1.
a
=6
2
3. 2c + 4 ! 5c + 7 = 0
5.
4
18
G=
9
5
7.
7k 2 12
!
=
14 14 14
2. 4 ! b = 3
(
4. !2 = !5 3d + 10
6. h +
2
=1
3
8. 0.3m = 2.1
9. Find equivalent fractions with a common denominator.
10. Subtract
13
42
4
7
48
© 2010 Cheryl Wilcox
13
7
!2
42
48
)
Free Pre-Algebra
11. Change the fractions to decimals.
Lesson 34 ! page 9
12. Change the decimals to fractions in lowest terms.
a.
67
100
a. 0.16
b.
67
99
b. 0.005
2
c.
3
c. 8.5
13. Find the circumference of a circle with radius 5.4 cm.
Round to the nearest tenth.
14. Find the area of a circle with radius 56.34 m. Round to
the nearest hundredth.
16. Find the volume of a sphere with radius 395 miles.
Round to the nearest whole number.
15. Simplify and then solve.
17. The gas mileage for Anton’s car is usually 28.8 mpg
(highway). If he has 19.2 gallons in his tank, how far can he
expect to drive on the highway before stopping for gas?
18. The circumference of a circle is 8.8 cm, measured to the
nearest tenth. What is the radius, to the nearest tenth?
19. Find the height of a rock dropped from the North Rim
into the Grand Canyon after 18.6 seconds. The equation is
h = !16t 2 + 8200 .
The elevation of the canyon floor is 2000 feet (the rock will
hit the ground at 2000 feet above sea level). Has the rock hit
ground yet?
© 2010 Cheryl Wilcox
0.25x ! 0.6x = 9.5 ! 8.075
Free Pre-Algebra
Lesson 34 ! page 10
Lesson 34: Algebra with Decimals
Homework 34A Answers
Solve the equations.
1.
2. 4 ! b = 3
a
=6
2
a
=6
2
a = 12
2•
!b + 4 = 3
!b = !1
b =1
a
= 6•2
2
(
3. 2c + 4 ! 5c + 7 = 0
4. !2 = !5 3d + 10
2c + !5c + 4 + 7 = 0
!3c + 11= 0
! 3c + 11! 11= 0 ! 11
!3c = !11 ! 3c / !3 = !11/ !3
11
c=
3
!15d ! 50 = !2
!15d = 48
16
d=!
5
5.
4
18
G=
9
5
6. h +
h+
9 4
18 9
• G=
•
4 9
5 4
2
G=
7.
81
10
2
2 2
2
=1
h + ! = 1!
3
3 3
3
3 2 1
h= ! =
3 3 3
3m = 21
m=7
7k ! 2 + 2 = 12 + 2
7k / 7 = 14 / 7
9. Find equivalent fractions with a common denominator.
10. Subtract
13
13
2 • 2 • 2 104
=
•
=
42 2 • 3 • 7 2 • 2 • 2 336
4
7
7
7
49
=
• =
48 2 • 2 • 2 • 2 • 3 7 336
© 2010 Cheryl Wilcox
! 15d ! 50 + 50 = !2 + 50
! 15d / !15 = 48 / !15
8. 0.3m = 2.1
7k 2 12
!
=
14 14 14
7k ! 2 = 12
7k = 14
k=2
)
2
=1
3
9
4
18
G=
9
5
!b + 4 ! 4 = 3! 4
3m / 3 = 21/ 3
13
7
!2
42
48
=4
104
49
55
!2
=2
336
336
236
Free Pre-Algebra
Lesson 34 ! page 11
11. Change the fractions to decimals.
12. Change the decimals to fractions in lowest terms.
a.
67
= 0.67
100
a. 0.16 =
b.
67
= 0.67
99
b. 0.005 =
c.
2
= 0.6
3
c. 8.5 = 8
13. Find the circumference of a circle with radius 5.4 cm.
Round to the nearest tenth.
C = 2!r = 2! (5.4 cm)
" 33.9 cm
16. Find the volume of a sphere with radius 395 miles.
Round to the nearest whole number.
(
)
4 3 4
!r = ! 395 mi
3
3
" 258,154,616 cubic miles
V=
3
17. The gas mileage for Anton’s car is usually 28.8 mpg
(highway). If he has 19.2 gallons in his tank, how far can he
expect to drive on the highway before stopping for gas?
E=
M
G
M
19.2
M
19.2 •
= 28.8 • 19.2
19.2
28.8 =
M
= 28.8
19.2
M = 552.96
16
4
=
100 25
5
1
=
1000 200
5
1
=8
10
2
14. Find the area of a circle with radius 56.34 m. Round to
the nearest hundredth.
A = !r 2 = ! (56.34 m)2
" 9972.03 m2
15. Simplify and then solve.
0.25x ! 0.6x = 9.5 ! 8.075
!0.35x = 1.425
! .35x / !.35 = 1.425 / !.35
x = !40.714285 " !40.714
18. The circumference of a circle is 8.8 cm, measured to the
nearest tenth. What is the radius, to the nearest tenth?
C = 2!r
2!r = 8.8
r = 1.4
8.8 = 2!r
2!r / (2! ) = 8.8 / (2! )
The radius is about 1.4 cm.
He can drive about 550 miles.
19. Find the height of a rock dropped from the North Rim
into the Grand Canyon after 18.6 seconds. The equation is
h = !16t 2 + 8200 .
The elevation of the canyon floor is 2000 feet (the rock will
hit the ground at 2000 feet above sea level). Has the rock hit
ground yet?
© 2010 Cheryl Wilcox
(
)
2
h = !16 18.6 + 8200
= 2664.64 feet
The rock has not yet hit the ground.
The elevation is 2664.64 feet.
Free Pre-Algebra
Lesson 34 ! page 12
Lesson 34: Algebra with Decimals
Homework 34B
Name _______________________________________
Solve the equations.
1.
z
= !3
5
3. 5x ! 7 + 7x ! 9 = 0
5.
5v
5
=
2 18
7.
5p 7 18
!
=
35 35 35
2. !4 ! y = 9
(
4. !10 = !5 9 ! 2w
6. t !
8
=7
9
8. 0.03n = 0.21
9. Find equivalent fractions with a common denominator.
10. Subtract
19
50
6
7
75
© 2010 Cheryl Wilcox
7
19
!4
75
50
)
Free Pre-Algebra
11. Change the fractions to decimals.
Lesson 34 ! page 13
12. Change the decimals to fractions in lowest terms.
a.
1
6
a. 0.9
b.
16
100
b. 0.09
16
c.
99
c. 0.099
13. Find the circumference of a circle with radius 15.2 cm.
Round to the nearest tenth.
14. Find the area of a circle with radius 6.88 m. Round to the
nearest hundredth.
16. Find the volume of a sphere with radius 7 miles. Round
to the nearest whole number.
15. Simplify and then solve.
17. The gas mileage for Farley’s car is usually 14.6 mpg
(highway). If he has 24.8 gallons in his tank, how far can he
expect to drive on the highway before stopping for gas?
18. The circumference of a circle is 2.8 cm, measured to the
nearest tenth. What is the radius, to the nearest tenth?
19. Find the height of a rock dropped from the North Rim
into the Grand Canyon after 19.7 seconds. The equation is
h = !16t 2 + 8200 .
The elevation of the canyon floor is 2000 feet (the rock will
hit the ground at 2000 feet above sea level). Has the rock hit
ground yet?
© 2010 Cheryl Wilcox
0.5x ! 1x = 76 ! 0.8