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Lesson 32 ! page 1
Lesson 32
Arithmetic with Decimals
Because decimals are a continuation of the place value system, decimal arithmetic is very similar to arithmetic with whole
numbers. In this lesson, we quickly review how to do each arithmetic operation with pencil and paper. However, decimal
arithmetic is most commonly done with a calculator, and that is the expectation of this text.
Addition and Subtraction
When adding or subtracting, align the numbers at the decimal point. This ensures the numbers in the same column have the
same place value. Use the same method as in adding whole numbers.
Example: Add or subtract as indicated. Check with your calculator.
0.03 + 2.9
5.08 ! 0.99
+
0.03
2.9
23.49 + 38.51
!
2.93
5.08
0.99
+
4.09
23.49
38.51
62.00
Notice that 0.99 is very close to 1. You
could subtract 1, then add 0.01 to get
the result in your head.
The number 62.00 is equal to just plain
62. Sometimes we leave the decimal
places filled with zeros because it
shows the level of accuracy with which
we began the problem.
Want further demos?
Addition: http://www.youtube.com/watch?v=GlJofAXeCHU
Subtraction: http://www.youtube.com/watch?v=io_Pxk13Luc&feature=channel
Multiplication
To multiply numbers with decimals, drop the decimal points, and multiply as with whole numbers. Then count the number of
digits after the decimal point in each of the two numbers you are multiplying. Add to find the number of digits after the
decimal point in your answer.
Example: Multiply.
0.03 ! 2.9
The first number has two digits after the decimal
point, and the second number has one digit. The
product will have three digits after the decimal
point.
29
!3
Now write the result with three digits after the decimal
point. The result of the multiplication goes all the way to
the right:
87
0.087
Want further demos?
Multiplication: http://www.youtube.com/watch?v=eI6Xp6pwhqc&feature=channel
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 32 ! page 2
Division
Moving the decimal point in the divisor allows us to locate it easily in the quotient.
Example: Divide.
0.85 ÷ 0.3
Start by writing a long division:
0.3 0.85
Change the number outside the
division symbol to a whole number. We
change 0.3 to 3 by moving the decimal
point one place right. We match the
movement of the decimal point in the
other number, so 0.85 becomes 8.5.
Now use long or short division. The
decimal point of the answer is
immediately above the decimal point of
the number inside the division symbol.
You can add zeros to the end of the
number inside the division symbol until
the decimal terminates or the repeating
pattern becomes clear.
2.833
3 8.500
6
2.5
2.4
0.10
0.09
0.010
?.???
3 8.500
0.009
3. 8.5
Video Demo of Divison:
http://www.youtube.com/watch?v=UiB1pB_5leU
Exponents
To evaluate expressions with exponents, convert to multiplication.
Example: Evaluate.
(0.4 )
(1.4 )
2
2
( )( )
= 0.4 0.4 = 0.16
(0.5)
= ( 0.5) ( 0.5) ( 0.5) = 0.125
3
( )( )
= 1.4 1.4 = 1.96
Compare to 42 = 16.
Compare to 142 = 196.
Compare to 53 = 125
(0.04 )
= ( 0.04 ) ( 0.04 ) = 0.0016
(1.04 )
= (1.04 ) (1.04 ) = 1.0816
(0.005)
= ( 0.005) ( 0.005) ( 0.005)
2
2
Compare to 1042 = 10,816
© 2010 Cheryl Wilcox
3
= 0.000000125
Free Pre-Algebra
Lesson 32 ! page 3
Using a Calculator Effectively
At this point in the course, you should already own a scientific calculator. If not, rush right out and get one immediately.
Simple calculators do operations in the order they are entered, and do not have a key for exponents. The main reason to
use a scientific rather than a simple four-function calculator is that scientific (and graphing) calculators use the correct order
of operations in evaluating long expressions.
Four-Function Calculator
Scientific Calculator
Graphing Calculator
Mainly for doing simple budgeting and
checkbook calculations.
Forever useful in all math and science
courses.
Commonly used in precalculus and
calculus courses. Necessary for
statistics.
Approximately $2 – $10.
Approximately $10 – $20.
Approximately $100 – $140.
Order of Operations
These problems are good practice in being adept with your calculator. Try to type in the expression in the order written,
including parentheses, and wait until the end to press enter. You will need to experiment with your calculator to see if you
need to press the multiply key between a number and parentheses, as in the example.
Example: Evaluate each expression.
(
)
0.7 0.4 + 7.88 With calculator:
(
0.7 0.4 + 7.88
)
With pencil and paper:
(
= 0.7 ( 8.28)
0.7 0.4 + 7.88
= 5.796
Notice that you don’t need the zero in the ones place when
entering decimals on the calculator. The calculator will put it
in for you.
© 2010 Cheryl Wilcox
)
Free Pre-Algebra
Lesson 32 ! page 4
Evaluating Algebraic Expressions
It will be a relief to finally be able to use decimals in formulas.
Example: Use a formula to solve the problem.
Convert 75.2ºF to Celsius, using the formula C =
.
F ! 32
1.8
(75.2) ! 32 = 24ºC
Find the perimeter and area of a rectangle with length
4.2 cm and width 9.5 cm.
Perimeter:
(
) (
)
)(
)
P = 2 4.2 cm + 2 9.5 cm = 27.4 cm
1.8
Keystrokes:
(
2
Area: A = 4.2 cm 9.5 cm = 39.9 cm
A boat traveled at 22.5 mph for 7.5 hours. What distance did
it travel?
(
)( )
d = 22.5 7.5 = 168.75 miles
The height (in feet) of an orange t seconds after tossing it up
is given by the equation h = !16t 2 + 10t + 6 .
Find the height after 0.9 seconds.
( )
2
( )
h = !16 0.9 + 10 0.9 + 6 = 2.04 ft
Rounding Decimals
This is another situation where our experience with whole numbers is valuable. Rounding a decimal number to a given place
value or a given number of decimal places is very similar to rounding whole numbers.
Example: Round the numbers as instructed.
Round 6.7291 to the nearest tenth.
Round 6.7291 to the nearest hundredth.
Find the tenths place: 6.7291
Find the hundredths place: 6.7291
The next digit is the trigger: 6.7291
Since the trigger is less than 5, we do not round up.
The next digit is the trigger: 6.7291
Since the trigger is greater than 5, we round up.
6.7
The tenths place is now the smallest place value.
© 2010 Cheryl Wilcox
6.73
The hundredths place is now the smallest place value.
Free Pre-Algebra
Lesson 32 ! page 5
Example: Round the numbers as instructed.
Round 7.3 to the nearest thousandth.
Round 0.54 to the nearest ten-thousandth.
Write out enough of the repeating digits to get past the
thousandths place:
0.54545454…
7.333333…
Locate the thousandths place and the trigger.
0.54545
7.3333
Round as usual. The trigger is less than 5.
0.5455
7.333
Example: Round the numbers as instructed.
Round 3.1412 to two decimal places.
Round 2.186 to three decimal places.
This means there should be two digits after the decimal point
in your rounded answer. The third digit is the trigger.
3.1412
2.186666
3.14
2.187
Example: Solve using a formula. Round answers as indicated.
A box has sides 5.5 cm, 3.6 cm, and 4.1 cm. Find the
volume of the box. Round to the nearest tenth.
Convert –38.5ºC to Fahrenheit, using the formula
F = 1.8C + 32 . Round to the nearest whole degree.
( )( )( )
(
)
V = 5.5 3.6 4.1 = 81.18
F = 1.8 !38.5 + 32 = –37.3
The volume is approximately 81.2 cm3
–38.5ºC is approximately –37ºF
!
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 32 ! page 6
Lesson 32: Arithmetic with Decimals
Worksheet
Name ______________________________________________
Do the decimal arithmetic with pen and pencil. Check with your calculator. Then convert the numbers to fractions or mixed
numbers and redo the arithmetic with fractions. Compare your results.
1. Add 0.45 + 2.1
Convert the numbers in the previous problem to fractions and add.
2. Subtract 8.09 ! 6.5
Convert the numbers in the previous problem to fractions and subtract.
( )( )
3. Multiply 7.1 0.7
Convert the numbers in the previous problem to fractions and multiply.
4. Divide 1.1÷ 0.3
Convert the numbers in the previous problem to fractions and divide.
5. $1,000,000 was split between 33 winners.
How much did each winner receive? Round to
the nearest penny.
A rectangle has length 0.03 cm and width 0.205 cm. Find the area and
perimeter. Round to the nearest thousandth.
Area:
Perimeter:
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 32 ! page 7
Lesson 32: Arithmetic with Decimals
Homework 32A
Name _________________________________________
1. Addition.
!3 + !8
0.3 + 0.8
0.03 + 0.08
3
8
+
10 10
Write a word problem to go with one of the above arithmetic problems.
2. Subtraction.
2!7
0.2 ! 0.7
0.02 ! 0.07
2
7
!
10 10
Write a word problem to go with one of the above arithmetic problems.
3. Multiplication.
(14 )(9)
(1.4 )(0.9)
! 4 $! 9 $
#" 110 &% #" 10 &%
(1.4 )(9)
Write a word problem to go with one of the above arithmetic problems.
4. Division.
74 / 3
7.4 / 0.3
74 3
÷
10 10
Write a word problem to go with one of the above arithmetic problems.
© 2010 Cheryl Wilcox
7.4 / 3
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Lesson 32 ! page 8
5. Compare the perimeters.
6. Compare the areas.
a. Find the perimeter of a rectangle with length 15 cm and
width 24 cm.
a. Find the area of a rectangle with length 15 cm and width
24 cm.
b. Find the perimeter of a rectangle with length 1.5 cm and
width 2.4 cm.
b. Find the area of a rectangle with length 1.5 cm and width
2.4 cm.
c. How many times would an ant have to walk around the
small rectangle to go the same distance as an ant who
walked once around the large rectangle?
c. How many copies of the small rectangle will fit inside the
large rectangle?
7. Compare the volumes.
Find the volume of a box with sides
20 cm, 30 cm, and 40 cm.
Find the volume of a box with sides
0.2 cm, 0.3 cm, and 0.4 cm.
How many of the small boxes will fit
into the big box?
8. Conversions.
How many inches is 23 cm? (1 inch = 2.54 cm)
How many cm is 23 inches? (1 inch = 2.54 cm)
Round to the nearest tenth.
Round to the nearest tenth.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 32 ! page 9
Lesson 32: Arithmetic with Decimals
Homework 32A Answers
1. Addition
!3 + !8 = !11
0.3 + 0.8 = 1.1
0.03 + 0.08 = 0.11
3
8
11
+
=
10 10 10
Write a word problem to go with one of the above arithmetic problems.
Sample: The tax on one item was $0.03 and on the other item was $0.08. What was the total
amount of tax?
2. Subtraction
2 ! 7 = !5
0.2 ! 0.7 = !0.5
0.02 ! 0.07 = !0.05
2
7
5
1
!
=!
=!
10 10
10
2
Write a word problem to go with one of the above arithmetic problems.
Sample: I have to pay my friend $7 and I have only $2. How much do I owe?
3. Multiplication
(14 )(9)
(1.4 )(0.9) = 1.26
= 126
! 4 $ ! 9 $ 126 63
#" 110 &% #" 10 &% = 100 = 50
(1.4 )(9) = 12.6
Write a word problem to go with one of the above arithmetic problems.
Sample: A rectangle has sides 1.4 cm and 0.9 cm. What is the area?
4. Division
74 / 3 = 24.6 = 24
2
3
7.4 / 0.3
= 24.6
74 3
2
÷
= 24
10 10
3
Write a word problem to go with one of the above arithmetic problems.
Sample: How many 0.3 cm slices can be cut from a 7.4 cm loaf?
© 2010 Cheryl Wilcox
7.4 / 3 = 2.46
Free Pre-Algebra
Lesson 32 ! page 10
5. Compare the perimeters.
6. Compare the areas.
a. Find the perimeter of a rectangle with length 15 cm and
width 24 cm.
a. Find the area of a rectangle with length 15 cm and width
24 cm.
(
)
( )( )
P = 2 15 + 24 = 78 cm
A = 15 24 = 360 cm2
b. Find the perimeter of a rectangle with length 1.5 cm and
width 2.4 cm.
(
b. Find the area of a rectangle with length 1.5 cm and width
2.4 cm.
)
( )( )
P = 2 1.5 + 2.4 = 7.8 cm
A = 1.5 2.4 = 3.60 cm2
c. How many times would an ant have to walk around the
small rectangle to go the same distance as an ant who
walked once around the large rectangle?
78 / 7.8 = 10
Ten times around the small rectangle equals
once around the large rectangle.
c. How many copies of the small rectangle will fit inside the
large rectangle?
360 / 3.6 = 100
One hundred copies of the small rectangle fit
inside the large rectangle.
7. Compare the volumes.
Find the volume of a box with sides
20 cm, 30 cm, and 40 cm.
Find the volume of a box with sides
0.2 cm, 0.3 cm, and 0.4 cm.
( )( )( )
( )( )( )
V = 20 30 40
V = 0.2 0.3 0.4
= 24000 cm3
= 0.024 cm3
How many of the small boxes will fit
into the big box?
24,000 / 0.024 = 1,000,000
One million small boxes will
fit inside the large box.
8. Conversions
How many inches is 23 cm? (1 inch = 2.54 cm)
How many cm is 23 inches? (1 inch = 2.54 cm)
Round to the nearest tenth.
Round to the nearest tenth.
23 cm
1 in
•
= 23 / 2.54 in
1
2.54 cm
23 in 2.54 cm
•
= 58.42 cm
1
1 in
! 9.1 inches
! 58.4 cm
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 32 ! page 11
Lesson 32: Arithmetic with Decimals
Homework 32B
Name _________________________________________
1. Addition
!4 + 9
!0.4 + 0.9
!4 + 90
!0.04 + 0.9
Write a word problem to go with one of the above arithmetic problems.
2. Subtraction
12 ! 8
1.2 ! 0.8
0.12 ! 0.08
0.12 ! 0.8
Write a word problem to go with one of the above arithmetic problems.
3. Multiplication
( 4 )( !22)
(0.4 )( !22)
(0.4 )( !2.2)
(0.4 )( !0.22)
Write a word problem to go with one of the above arithmetic problems.
4. Division
48 ÷ 4.5
4.8 ÷ 4.5
48 ÷ 0.45
Write a word problem to go with one of the above arithmetic problems.
© 2010 Cheryl Wilcox
480 ÷ 45
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Lesson 32 ! page 12
5. Compare the perimeters.
6. Compare the areas.
a. Find the perimeter of a rectangle with length 40 cm and
width 50 cm.
a. Find the area of a rectangle with length 40 cm and width
50 cm.
b. Find the perimeter of a rectangle with length 0.4 cm and
width 0.5 cm.
b. Find the area of a rectangle with length 0.4 cm and width
0.5 cm.
c. How many times would an ant have to walk around the
small rectangle to go the same distance as an ant who
walked once around the large rectangle?
c. How many copies of the small rectangle will fit inside the
large rectangle?
7. Compare the volumes.
Find the volume of a box with sides
4 cm, 5 cm, and 6 cm.
Find the volume of a box with sides
0.4 cm, 0.5 cm, and 0.6 cm.
How many of the small boxes will fit
into the big box?
8. Conversions
How many inches is 95 cm? (1 inch = 2.54 cm)
How many cm is 95 inches? (1 inch = 2.54 cm)
Round to the nearest tenth.
Round to the nearest tenth.
© 2010 Cheryl Wilcox