# Decimals - DOC

Document Sample

```					               Decimals

Quick!   Pick a number between 0 and 1.

1
Place Value and Naming

Hundred thousandths
Hundred thousands

Ten thousandths
Ten thousands

Ten millionths
Thousandths
Ten millions

Hundredths
Thousands

Hundreds

Millionths
Decimal
Millions

Tenths
Ones
Tens
5      1        .        3          4             5

The number in the chart above is read “fifty-one and three hundred forty-five thousandths.”
Read the number before the decimal.
Say “and.” for the decimal.
Read the number following the decimal.
Add the word for the place value of the last digit.

0.04 is read 4 hundredths.

3.00013 is read "three and thirteen hundred thousandths."

"Fourteen and five hundred twelve millionths" must end on the millionths place, 14.000512

Whole numbers have a decimal at the end. 23 is 23.
The 3 is in the ones place and the 2 is in the tens place.

Practice: Write the following in words.
Use the chart above to write in the number. Find the place value of the last digit.
a) 2.30007                           0.034                             21.000034
b) 12.00017                          1.0041                            0.0000074
c) 1.0101                           123.321                           0.12345

Practice: Write the following numbers.
d) two and fifteen ten-thousandths                                      eighty two ten thousandths
e) eleven and eleven hundredths                                     three hundred twelve millionths
f) forty-five millionths                            seven and seven hundred seven ten thousandths
g) 2 tenths                            One hundred one and ten thousand one hundred one millionths

2
In money, \$23.45, the 2 is worth \$20 and the 4 is worth 4 tenths of a dollar.

0.02345 the 3 represents 3/1000 or three thousandths. The 5 represents 5/100000 or five hundred
thousandths.

Practice: State the value of the specified digit in the following number 23.4516879
4                                   6                                    5
2                                   7                                    9

3
Rounding

A dress costs \$73. Is 73 closer to \$70 or closer to \$80?
Of course it is closer to \$70.
This is an example of rounding.

What about \$97? Is 97 closer to \$90 or \$100?                    \$97 is closer to \$100.

Example: Round 168 to the nearest hundred.
Underline the hundreds position. 168.
Look to the right. Is the number right next door, the six, big enough to increase the underlined
position from 1 to a 2?

Yes. 168 is rounded to 200.     This says 168 is closer to 200 than 100.

If the digit just to the right of the position to be rounded is five or more, round up. If the digit is less
than five the digit to be rounded stays the same. All place values after the rounded digit are filled with
zeros.

Example: Round 3,284 to the nearest hundred.
Underline the hundreds place. 3,284
Look to the right. Is eight large enough to increase the two to a three? Yes.
3,284 rounded to the nearest hundred is 3,300.

The numbers after the hundreds place change to zeros.

Round 3,284 to the nearest thousand. Underline the thousands place. 3,284. Look at the hundreds
place. Is two large enough to change the three into a four? No. 3,284 rounded to the nearest
thousand is 3,000.

Practice: Round the following to the nearest ten.
a) 345               38                   52                      183                    189
b) 295               97                   92                      186                    1345

Round these to the nearest hundred
c) 432              584                     2346                  3586                   6745
d) 1693             2943                    134,565               6753                   6890

When rounding decimal numbers, the process is the same. Underline the desired place value. Look at
the digit to the right of the specified place value to decide if you increase or if you keep the same the
digit in the place value.

Example: Round 3.48294323 to the nearest hundredth.
 Underline the place value asked for. 3.48294323
 Look at the number next door, the 2. Is two large enough to increase the 8? No.
 3.48 Drop the rest of the number.

Practice: Round the following to the nearest tenth.
a) 0.345             0.38                 0.52                    0.183                  0.189
b) 2.95              9.72456              9.2764                  1.86                   1.8345

4
Round these to the nearest hundredth
c) 0.432            0.584               0.2346        0.3586      0.6745
d) 1.693            2.943               13.4565       6.753       6.890

Round these to the nearest thousandth
e) 0.43243          0.58465             0.234676      0.358635    0.67455
f) 1.693245         2.942453            13.45653256   6.7533568   6.89068

5
Add, Subtract and Compare

Suppose you go to a store and purchase a \$23 item and a \$7 item. What would you  23
say if the clerk added the items this way?                                      7

Would you pay \$93?                                                                   93
Of course not!

The numbers were not lined up correctly. The 7 should be in the one's place.
When adding, ones should be added to ones, tens to tens, and hundreds to hundreds.
The same is true for tenths, hundredths, and thousandths.

To insure that like place values are added, line up the decimal first.
Follow the example: 2.3 + 4 + 5.023 Even whole numbers have a decimal.
The 4 has a decimal after it.

2.3 + 4 + 5.023
1) Line up the decimal.           2.3              2) Fill in zeros.        2.300
4.                                        4.000
5.023                                     5.023

Practice: Add neatly on paper. Then check your work with a calculator.

a) 3.4 + 2.1 + 7.2=       2.35 + 1.23=             31.41 + 53.27=          0.42 + 0.53 + 0.03=

b) 5 + 3.1=               24.53 + 31=              2.35 + 7 + 3.41=        3.21 + 42.53 +0.13=

c) 3.1 + 0.234 +1.23=     0.4 + 2.301 + 5=         0.321 + 0.24 + 0.3=     23.5 + 2.03 + 4=

d) 234.11 + 3.257=        4.2 + 0.345 + 743=       1.234 + 43.21 +234.1=   1.234 +12.34 + 123.4=

6
Carrying or Regrouping

1       1   When you add 15 and 27, the 7 and 5 make 12.
15     15      15 The one is placed in the tens column to be added to other tens.
27   27      27 Write the carried one smaller than the original numbers to be added.
2    42

Practice: Write the following addition problems vertically, lining up the decimal. Add. Check your work
with a calculator.
a) 4 + 5 + 8=              5 + 9 + 2=                 0.2 + 0.8 + 0.5=           0.4 + 0.7=

b) 0.1 + 0.8 + 0.5=        23 + 12 =                15 + 97=                  1.4 + 1.8=

c) 98 + 98 =               77 + 18 =                5.8 + 8.9=                2.9 + 3.1=

d) 5.7 + 3.4 =             1.3 + 7.7 =              9 + 9.6+ 2.6 =            2.3 + 0.8 +5.4=

e) 14 + 8.2 + 9.9=         555 + 862 + 34 =         585 + 813 + 394 =         1.23+98.7+5.4=

f) 37.25+2.893=            35.67+826.9=             2.76+427.5+34=            1.279+65.02+540.432=

g) 2.458+108.53=           154.7+9.876+12=          5.427+1.9082+0.743=       4+6+2.53=

7
Measuring Length Using Centimeters

The line is 4.2 cm. Start the ruler at zero. The tiny
spaces between the larger numbered lines are
millimeters. They are one tenth of a centimeter.
Notice the line is 4 whole centimeters and two
tenths of centimeter.

Each centimeter has 10 smaller divisions. These are millimeters. 3.4 centimeters or 3.4 cm is the same
as 3 centimeters and 4 millimeters.

Practice: Measure the following lines and items in centimeters.
a)

b)

Measure each side of the following shapes. Add the lengths of the sides. The PERIMETER
is the total length around a flat shape.

a)

Rectangle
Parallelogram                Right triangle
b)

Pentagon
Isosceles Triangle           Regular hexagon
c)

Trapezoid                         Acute Triangle             Obtuse Triangle

8
Word Problems Involving Addition

a) A garden has a border of large white                Nancy made a frame for her son’s art work. It
rocks. The garden is 3.4 meters by 2.85                was 7.05 inches by 15.23 inches. How long
meters. How long is the border all the way             was the framing material Nancy used? (Ignore
around?                                                waste)

b) Jan put three packages on a scale.                  Three parts are to be welded. They are 3.45
Individually they weigh 4.2 kilograms, 2.37            in., 7.8 in., and .98 in. in length. What will be
kilograms, and .45 kilograms. What is on               the final length?
the scale display?

c) In 1990 there were 359.9 thousand                   The trip to the outlet store is 5.3 miles. From
people in Mytown. By 1996 there were 8.6               there we will go to the campus library that is
thousand more people in town. How many                 3.02 miles. From the library it is a 2 mile
people lived in Mytown in 1996?                        straight ride home. How far will we drive?

d) The balance in the check book is \$130.              A new couch is on sale for \$1234.99. The love
Sheila deposited a check for \$24.51 and                seat is \$608.49, the coffee table is \$300. What
\$12.34 in cash. What is the new balance?               is the total for the set?

e) Your boss requested a breakdown of man              The first track on my favorite CD is 5.03
hours spent to date for the West Coast                 minute long. The next three are 3.4, 5.45, and
project. 30.52 hours were spent on needs               3 minutes. How much playing time is there
assessment, 52.8 hours on the proposal, 25             for the first four songs.
hours planning, and 120 hours in
construction. What is the total of man-
hours?

f) An apple cost \$0.52, an orange is \$0.70,            The company made \$4.56 million more this
and a banana is \$0.38. What is the total               year than last year. they made \$12.8 million
cost?                                                  last year. How much did they make this year?

g) Write two word problems that use addition of decimals to solve. One should require the addition of
three or more numbers.

9
Equivalent Decimals

034 is the same as 34. The zero in the hundreds place isn't necessary. The number still has no
hundredths, 3 tens and 4 ones.
1.5 is the same as 1.50 The zero at the end is not necessary, but helps in calculations. The number
still has 1 one, 5 tenths, and no hundredths.

You may add as many zeros left of digits that are before a decimal. You may also add as many zeros
as you want right of the digits after the decimal.

000032.456000 = 032.456 = 32.456 = 32.4560000
These all have 3 tens,, 2 ones, 4 tenths, 5 hundredths and 6 thousandths.

0.42 is the same as .42, however if there is no whole part (ones, tens, etc..) in a number a single zero
in front of the decimal is proper.

Circle the pair of equivalent decimal numbers in each row.
a) 00.432000          04.32               000.4320              0432                  43.200

b) 50.0              5.0                   50                   0.5                   05

c) 987.5400          9.8754                9875.4               987.54                98.754

d) 045.38            .4538                 0.4538               04.538                04538

Comparing or Ordering

1) Line up the decimal
2) Fill in zeros

These steps insure tenths will be compared to tenths, hundredths to hundredths, thousandths to
thousandths and so on.

Example: Put these numbers in order from the largest to the smallest.

0.32, 0.203, 0.03, 0.3

1) Line up the        0.32
decimal               0.203
0.03
0.3

2) Fill in zeros      0.320        three hundred twenty thousandths
0.203        two hundred three thousandths
0.030        thirty thousandths
0.300        three hundred thousandths

Now compare three hundred twenty thousandths to two hundred three thousandths, thirty
thousandths, and three hundred thousandths.

10
0.32 is the largest, then 0.3, 0.203, and finally 0.03

Put the following in order form largest to smallest.
a) 0.58, 0.825, 0.08, 0.5              3.14, 0.009, 0.095, 0.01   0.07, 0.6, 0.008, 5
b) 0.13, 0.059, 0.4, .34               0.05, 2.09, 2.010, 0.063   .3, 0.049, 0.501, 0.0601

Put the following in order form smallest to largest.
a) 0.045, 0.54, 0.504, 0.405          9.5, 0.59, 0.95, 0.509      0.164, 0.709, 0.482, 256
b) 0.09, 0.3, 0.05, 0.010             0.395, 0.054, 0.73, 0.98    0.72, 0.295, 476, 0.163

11
The symbol < is read “less than”                   5<12 Five is less than twelve.

and > is read “greater than.”                      295 > 34 two hundred ninety five is greater than
thirty four.

“The alligator turns to eat the bigger number” is an easy way to remember which way to write the
symbol.

854                   342 and 12                 987

Insert the symbols <,> or = between these pairs of numbers
a) 12_____62             1.23______1.3             4.008_______4.03         1987______1978
b) 1.2_____0.62          0.24______0.204           0.509_______0.6          1.582_____1.59

12
Subtraction with Borrowing
234 - 87 line up the numbers.
534        7 is bigger than 4 so we borrow a ten from                  5 23 14              7 from 14 is 7.          5 23 14
87        the tens place, change it to ten ones and
add it to the 4. We now have 14 in the                       8 7                                          8 7
ones place and a 2 in the tens place.                                                                              7

4
5   12
3 14   8 tens is larger than 2 tens so borrow from       4
5   12
3 14    Complete the
the hundreds place. This leaves 4 in the                               subtraction.
         8 7        hundreds place and 12 in the tens place.                  8 7
7                                                              44 7

Note the borrow and carry numbers are written smaller than the numbers in the original problem.
This avoids confusion. This process is the same in decimal subtraction.

To insure tenths are subtracted from tenths, hundredths from hundredths and so on,
1) Line up the decimal before subtracting.
2) Fill in Zeros

4             9
5.000                                        5 . 90 90 1 0
5                                                                     1  1
Example: 5  0.343 Step one                        Step two       0.343       Borrowing is required.           0. 3 4 3
0.343
4.657                                        4. 6 5 7
Notice the decimal is after the 5. The decimal stays in line for the answer.

Practice:
a) 537 - 115                      493 - 352             345 - 133                      943 - 133                      739 - 606

b) 371 - 286                      493 - 395             345 - 258                      943 - 789                      739 - 359

c) 3.1 - 2                       3.57 - 3.15           6.09 - 5.06                    0.27 - 0.07                   45.8 - 32.6

d) 5. 09 - 3.99                  7.36 - 6.39            530 - 52.9                   0.968 - 0.899                     5.2 - 4.7

c) 12 -0.398                     6.32 - 4.11           52.1 - 31.0                     46 - 0.278                   3.62 - 2.24

f) 2.62 - 1.78                     93 - 5.1             2.5 - .23                    0.476 - 0.253                0.924 -0.646

g) 6.5 - 2.4                     4.23 - 2.34           34.5 - 25.6                    3.51 - 3.49                      432 - 78

13
Word Problems Involving Subtraction.

a) Tiny Car company made 5.6 million last             A tree was 13.4 meters tall in 1999. In 2002 it
year. This year they made 12.3 million.               grew to 14.1 meters.
How much more did Tiny Car company                    How much did the tree grow?
make this year?

b) Randy drives 34.5 miles to grandma's               A triangular frame is 36.8 inches around. Two
house. His brother, Mike, drives 72.3 miles.          sides measure 9.45 inches and 13 inches. What
How much farther does Mike drive?                     is the measure of the last side?

c) A wool scarf that was 3.2 feet long                If the price of a gallon of gasoline rose from
before the wash is now 2.8 feet long. How             \$0.98 to \$1.34, how much did the price of a
much did the scarf shrink?                            gallon rise?

d) David had the flue for Thanksgiving                Eight swim lessons cost \$12.85 for members of
weekend. His temperature was 103.2 at                 the club and \$16.50 for non-members. How
the highest. His normal temperature is                much more do non-members pay for eight
98.6. How much higher was his                         lessons.
temperature during his fever?

e) Your gross pay is \$342.43. Federal tax             Zech was 21 pounds at three months. He was
is \$45.69. State tax is \$9.45, and FICA is            7.3 pounds at birth. How much weight did he
\$19. What is your take-home pay?                      gain in three months?

f) A rectangle is 32 feet wide with a                 A rectangle is 0.258 feet wide with a perimeter
perimeter of 378 feet. What is the length?            of 0.756 feet. What is the length?

g) Write two word problems that use subtraction of decimals to solve.

14
Multiplication
1) Multiply
2) Count place values
3) Move the decimal left that number of spaces.
Example: 2.05 x 0.003

2.05
0.003                                                                                           2.05
Five place values, so move the decimal five places left. Add necessary zeros.     0.003
615
0.00615

Steps: Examples:
Ignore the                            3.245                                                              0.003
decimal     3.245 x 0.02 is rewritten                                     5 x 0.003 can be rewritten
 0.02                                                                 5

Multiply          3.245                                                                                  0.003
 0.02                                                                                  5
6490                                                                                     15

Move the          3.245
decimal
0.003
 0.02     Move the decimal five
.06490                                           5
places. Fill in zeros as needed.                          Move the decimal three places.

Then drop the ending zero and put
.015
the “polite zero” in the front.                                                        T
0.0649                                          Then put a zero in the ones place. 0.015

Why: Multiplying by 0.3 is the same as multiplying by 3 and dividing by 10. When you divide by 10
you move the decimal one space to the left. So multiplying by 0.0005 you will move the decimal four
spaces left.

Practice: Write the multiplication vertically. After multiplying, move the decimal the
correct number of places. Fill in zeros as needed.
a)      0.3 x 0.5         0.03 x 0.5         0.03 x 0.05       0.3 x 0.005     0.0003 x 0.5

b)           4 x 0.5             4 x 0.05             4 x 1.05         4.01 x 0.05             4 x 5.1

c)           0.23 x 6          0.01 x 3.14           5.3 x .031        .002 x .103           5.8 x .0006

d)         1.50 x 0.64          6.3 x 0.11           2.1 x 31.0          0.4 x 2.7           3.62 x 2.24

e)         0.62 x 0.007          9.3 x .51           2.5 x .23         0.47 x 0.25           0.94 x 0.46

15
f)   6.5 x 2.4   4.3 x 2.34   34.5 x 20.6   3.01 x 3.4   0.2 x 0.78

16
Word Problems Involving Multiplication

a) 1 inch = 2.54 centimeters. How many             How many meters are there in 567 yards? 1yard
centimeters are there in 56 inches?                = 0.914 meters.

b) At \$6.42 a cubic foot, how much does            What is the weight of 36 cubic feet of water? A
8.25 cubic feet of concrete cost?                  cubic foot of water weighs 62.5 pounds

c) Kathy worked 5.4 hours of overtime at           The television costs \$0.08 per hour to run. How
\$25.42 per hour. How much did she                  much will it cost to watch 3.45 hours?
make?

d) The baby weighs 49 pounds. The                  The average speed of the Enterprise is 521 light
weight is to be charted in kilograms.              years per hour. How far did the ship fly in 0.325
What would you chart?                              hours.?
One pound is equal to 0.45 kilograms.

e) One mile is equal to 1.6 kilometers.            A garden is 15.4 m by 13.02 meters. What is the
How many kilometers are there in 73                area? Hint:(Multiply the lengths of the sides to find
miles?                                             the area of a rectangle)

f) Find the area of a rectangle with width         A room is 37.75 meters by 20.5 meters. What is
0.25 meters and length 0.307 meters.               the area?

g) Write two word problems that use multiplication of decimals to solve.

17
Division
1) Move the decimal by the “house” if it’s not already
there.                                                         Remember F.I.T.
2) Match the move inside the “house”
3) The decimal in the answer is straight above the             The First number using ÷ is the same as
decimal inside the house.                                      the Inside number when using the house
4) Divide normally.                                            and the same as
the Top number when using the fraction bar.
Example: 80.64 ÷ 2.56
31.5
2.56 80.64                                                    2.56 . 80.64 .00
Move each decimal 2 places.

Practice.
1.258 ÷ 0.8           1.258 ÷ 0.08        125.8 ÷ 0.08             1258 ÷ 0.08             1.258 ÷ 0.008

.7
Example: 3.5 ÷ 5                                                               5 3.5

There is no need to move the decimal because the decimal in the
five is after the five and next to the house.

Practice.
1.258 ÷ 8             18.571 ÷ 7          0.7 ÷ 20                 28.5 ÷ 12               0.639 ÷ 15

Add as many zeros as you need.
Round the answer to the desired accuracy.

Example: 1.3 ÷ 0.07 Round the answer to the nearest hundredth.

To get an answer rounded to the nearest hundredth, you will work the problem to the thousandth
place to know if you should round up or not.

1               18           18.5               18.57               18.571
0.07 1.3       0.07 1.30      0.07 1.30     0.07 1.300         0.07 1.3000         0.07 1.30000
The answer is 18.57 rounded to the nearest hundredth.

Practice. Round each answer to the nearest hundredth.
a) 1.5 ÷ 0.64           63.2 ÷ 4.11           52.1÷31                  4.64 ÷27.8                  3.82 ÷ 8.24

b) 2.62 ÷ 1.78              0.93 ÷51             2.5 ÷ .23           0.476 ÷ 0.253               0.924 ÷0.646

c) 6.5 ÷ 2.4               4.23 ÷ 2.34           4.5 ÷ 25.6            3.51÷3.49                   4.32÷0.078

18
Word Problems with Division of Decimals

The number inside the house is the total, the thing being split, sorted, cut, shared, divided, or
distributed.
The number outside the house is either how many pieces or how big those pieces should be.

How big                            How many
How many Total                      How big Total

Example: Tony has 5.8 kg of coffee. He will put them in 8 bags. How much will be in each bag?

Total 5.8 kg of coffee
.725
How many? 8 bags.            8 5.800
Tony will put 0.725 kg of coffee in each bag.

Example: Randy has 5.8 kg coffee. He wants 0.25 kg bags. How many bags will he need?

Total 5.8 kg
23 .2
How big? 0.25 kg bags.     0.25 5.80 0
Randy will have 23 full bags and one bag with two tenths of a kilogram.
He will need a total of 24 bags.

a) There are 5 people on the elevator.                     There are 2.54 centimeters in one inch.
Together they weigh 925.98 lbs. Find the                   How many inches are there in 51.78
average weight.                                            centimeters?

b) What was its average speed in miles per                 There are 1.6 kilometers in a mile. How
hour if a plane flew 1856.4 miles in 5.2 hours?            many miles are there in 98.7 kilometers?

c) Gerry makes \$7.58 an hour. He grossed                   What is the average number of miles per
\$306.99 last week. How many hours did he                   day Daniel drove on a 1345 mile trip done in
work?                                                      6 ½ days? (hint change the ½ to .5.)

d) Larry works 15 days for a total of 532.25               My bottle of medicine contains 21.5 ounces.
hours. How many hours did he average per                   I take 1.75 ounces each dose. How many
day?                                                       doses are in the bottle?

e) If 5.3 pounds of nails cost \$16.96 what is              2.5 yards of polar fleece cost \$26 how much
the price of one pound of nails?                           does one yard cost?

f) Last year the precipitation was 3.5 inches in           The readings on the dial were 3.4, 5, 2.75,
December, 2.5 inches in January and 3 inches               and 3.61. What is the average of the
in February. What was the average                          readings?
precipitation for last winter?
(Hint: Add the three numbers and divide by 3.)

19
g) Write two word problems that use division of decimals to solve.

20
Estimation

Estimation is doing math in your head. You get an answer that is not exact.
1) Round the numbers in the question.
2) Work the problem with the rounded numbers.

Example: Estimate 3.14 x 4.871
Round 3.14 is close to 3 and 4.871 is close to 5
Work the problem 3 x 5 = 15
3.14 x 4.871 is approximately 15.

Estimating division answers is a bit more complicated. Instead of just rounding the numbers, use
numbers that are close to the numbers in question, but that divide evenly.

Practice: Estimate each of the following.
a)        7.3 x 9.5          5.03 - 0.95           78.3 + 8.05    14.8 ÷ 2.872          12.8 x 1.5

b)        4 - 0.8             46 ÷ 4.3               4 + 1.05      4.01 x 7.05           4 x 5.1

c)        2.3 x 6           11.78 - 3.14           35.3 ÷ 3.031      2 x 10.3          5.8 + 6.87

d)     7.50 ÷ 1.364          6.3 + 0.11            2.1 x 31.05      5.4 x 4.7         83.62 - 22.24

e)    15.62 + 8.007         9.3 ÷ 2.518             2.5 x 8.23     0.47 ÷ 0.25         9.94 - 6.46

f)       6.5 x 2.4           4.3 ÷ 2.34            34.5 + 20.6      8.01 - 3.4          0.9 x 0.78

21
To change a fraction to a decimal, divide.
The fraction bar means divide. 3/8 says "three divided by eight."          Divide the 3 by the 8
0.            0.3      0.37           0.375
83        8 3.0         8 3.0    8 3.00         8 3.000

3
To change 5     to a decimal, save the whole number “5” and work only with the ¾. Divide 3 by 4.
4
0.75
3
4 3.00 Then put the five back in position. 5 =5.75.
4

Practice: Write the following as decimal numbers. Round to the thousandths place.
a) 2/3                              4 1/8                            5
/4

b) ¾                                    ½                              3 3/8

c) 1/3                                  2
/5                           2 3/10

d) 5/12                                 3
/16                          2
/7

e) 1/4                                  5 5/8                          7
/8

f) 4/3                                  3
/5                           9
/8

g) 3/25                                 6
/7                           4
/5

To change a decimal to a fraction, read the number, write it as a fraction, then reduce.

Read the number. Then write the number using the place value and reduce.

Example: Change 0.25 to a fraction.
0.25 is read “twenty five hundredths.”
25
/ 100= 1/4

45             9
Example: 3.45 Save the 3. Read .45 as forty-five hundredths. Write       . Reduce to    .
100             20
The result is 3 9/20.

Practice: Write the following as fraction numbers.
a) 0.4                      2.01                          3.75

b) 3.05                         0.25                      7.8

c) 0.3                          4.1                       8.005

d) 0.92                         2.025                     0.375

e) 1.0012                       3.375                     25.25

22
f) 0.875   0.625    0.3

g) 2.005   1.0002   5.02

23
Order of Operations

Look at both of the problems. Notice the difference in the way they are solved.
48  6  2 The only difference in the way these problems were done is the               48  6  2
order the operations were performed.
 8 2                                                                                   48  12
 16           The one on the left is correct.                                          4
Multiplication and division are always done left to right.
Remember multiplication is commutative and associative, but division is not. You can do problems that
contain only multiplication in any order, but if division is in the problem, then the order is important.

144  6  2  3  4
81  3  5                 4  21  2
 24  2  3  4
 27  5                    84  2
 48  3  4
 135                       42
 16  4  64
Practice:
a) 5÷10x200=                       45÷3x2=                       8÷4x7=                   6÷3x10÷5x9=

b) 4.5x2÷5=                       510÷5÷12=                      8÷4x7=                        42x3÷9=

c) 4x3÷5x24=                       2÷5x8=                        2x5÷8=                          2x5x8=

Addition and subtraction work the same way. Subtraction isn’t commutative. Remember to think “add
the opposite” when subtracting, but do it left to right.

Practice:
d) 9+3-5=                          9-3+5=                        45-3+2=                      12-6-3+7=

e) 534-83+29=                  3.4 - 1.2 - 0.65=          5.34 - 0.24 + 4.999=     5.34 + 0.24 - 4.999=

Multiplication and division are always done before addition and subtraction. Write each step out
completely under the previous step.

We use three ways to indicate multiplication. 3x4, 3(4) and 3·4 all mean multiplication.
5  5  2  8  Multiplication before          85  7 8               8  4 6  3  5
subtraction.
25  16                                                                 2 6   3  5
Note: Write the new            40  56 
9               problem after multiplying.                                12  3  5
96
9  5  14
Practice:
f) 45+3x2=                      20-6÷3x9=                    2.3÷4+5=                    5÷8 - 0.003=

g) 4·5+3·2                        4÷5-3x2=                       12÷4+5(2)=              8 - 2.3÷4 + 5=

24
There is a mistake in each of the following problems. Discover what was done incorrectly.
12  4  2                          9  12  3                                    15  3  5 
h)                    4 is correct.
8  2  16                           9  36           2.25 is correct.             15  8        17 is correct.
1
4                                         7
Multiply before subtraction

Exponents are done before multiplication and division.
9  23  7 2    43  2  32  2  5  Note in the examples, the                         An exponent is a way to show
exponents are done and                               repeated multiplication.
9  8  49      64  2  9  2  5      the rest of the problem is
72  49         32  18  5              written down.                                         34 means 3x3x3x3=81

23                     14  5  19                  If you scratch to the side                       52 means 5x5=25
and skip writing all steps,
you will make mistakes.                   25 means 2x2x2x2x2=32

Practice:
i) 52=                23=          32=              33=           25=              92=                43=          53=

j) 1.22=         0.52=            3.43=            0.052=        0.43=            1.52=             1.53=       6.122=

k) 23 + 52=                              25 - 42=                       54 - 33=                              23 x 34 =

l) 42 + 5 x 23=                       52 + 3 x 2 2 =              23 - 6 ÷ 3 x 32=                   50 ÷ 2 + 5 x 22=

m) 52(2) - 3 x 23=                    152 - 3 x 52 =

The first thing always done is parenthesis or other grouping symbols.
If there are nested grouping symbols, work from the inside out.
4  2  71  3  8  6    Work the inside parenthesis first.


4  2 71  3 14         The next set of parenthesis has two operations inside.
4  2  71  42           Always do multiplication before subtraction.
Finish the inside of the second parenthesis.
4  2  29  
4  58  62                 Multiply before subtracting.

Any symbol      that separates the problem into parts acts like a parenthesis.
65  8  7        The division bar groups the operations.

23  5
65  56           In the numerator the multiplication is done before subtraction.

85
In the denominator the exponent is done first.
9
3
3                 Finally the division is done.

25
To remember the order of operations use the mnemonic devise Please Excuse My Dear Aunt Sally.
Please       Parenthesis and other grouping symbols. 34  5 = The subtraction is done first
( ), { }, [ ],   ,                            3(-1)=-3          because it is in the
parenthesis.
Excuse           Exponents                                     144 - 43=
x2                                            The exponent is done before the subtraction.
144 - 64=80
My Dear          Multiply and Divide from left to right.        4  6  8  24  8  3
3x5, 3 5 and 3(5) all mean multiply.          72  3  4  24  4  96
12
12  4 ,      , and 4 12 are all the same
4
division.
Aunt Sally       Add and Subtract from left to right.          7 - 5+8=2+8=10

Practice: If necessary round each answer to the nearest thousandth.
a) 25 - 8(3+2)=                                             5 + 8( 3+2)=                                     30÷6x18=

b) 18 - 8(4 - 2)=                                               81 ÷ 6+3(7 - 4)=                              3(3) + 9(7-5)=

c) 7 – 3(2) + 5(14-3)=                                         12 - 10 + 89 – 72=                       3.4 – 1.7 + 0.9 + 7.2

d) 2(3.14)(5)2 + 2(3.14)(5)(7)                                  0.2 +0.5(5 - 0.6)=                          5 - 0.34(4 + 2)=

7  4  3                                                16  12  5                           4  32  6
3                                                               4
 8(2  5) =
e)               =                                                            =                             5
50  6  3                                                   25  6(3)

30   60  56                                                                                           82   5 
2
=
f)   8                  =                                          6 5
2      2
14  9                                                                =
7  18  3  5 

5  82  32                                                                                              5  38  32 
g)                 =                                              9(28)                                                      =
9                                                             32 =                                       9
5

9(8)                                                    0.3 + 0.81(8.1 - 2.435)=                      7.1(0.5) - 3÷1.2=
h)         32 =
5

26
Mean, Median, Mode, Range

Mean, median and mode are three kinds of averages. They can be the same answer for a particular
data set or they can be different. When you read an article in a magazine and it says “the average…”,
it is helpful to know what kind of average is being talked about.

Example: A family has children aged 3, 4, 5, 3, 6. find the mean, median and mode.

Mode: the number that appears the most often.

3 appears twice the other numbers only once. 3 is the mode. A list can have more than one mode or
no mode at all.

Median: The number in the middle of the list.

First, put the numbers in order.         3, 3, 4, 5, 6

The number in the middle is 4. 4 is the median.
If a list has an even number of elements, the median is the mean of the middle two numbers.

Mean: Mean is the kind of average you have used before.

1) Add the numbers.                 3+ 3+ 4+ 5+ 6 = 21

2) Divide by how many numbers were in the list. There are 5 numbers in the list.

21÷5 = 4.2

Practice:
a) Find the mean, median, and mode of the following set: 7, 5, 4, 6, 3, 8, 2, 5, 4, 6, 4, 7, 6.

b) Find the mean, median, and mode of the following: 2, 2, 3, 3, 4.

c) Find the mean, median, and mode of the following: 2.34, 12.3, 3.4, 4, 2.1, 5.32.

d) Find the mean, median, and mode of the following: 6, 6, 6, 6, 4, 4, 15, 30

e) Find the mean, median, and mode of the following: 2.101, 3.5, 1.2, 3, 3.6

f) Jack has taken 4 of 5 GED tests with the following scores: Math 480, Literature 430, Science 460,
Social studies 400. He needs a mean of 450 to pass. What score will he need to get on his last test to
have a mean of 450? What is the median and mode of the scores?

g) A class has 15 students. The mean age of the class is 29 years. The mode is 21 with 3 students at
that age. Two students are 18. Others are aged 16, 19, 27, 31, 35, 37, 40, 40, and 54. What is the
age of the last student? What is the median?

27
Mean, median, and mode help describe the center of the data.

Look at one student’s test scores: 40, 50, 60, 70, 70, 70, 80, 90, 100.
Compare to another student’s scores: 70, 70, 70, 70, 70, 70, 70, 70, 70, 70.

The mean, median, and mode are the same, but just by looking it can be seen the data sets are very
different.

Range tells how spread out the data is. In the first list, the student scores are spread out. While the
scores in second list are all the same.

The range is found by subtracting the smallest from the largest number in the list. The range in the
first set of student scores is 100-40=60 the range for the second set is 0.

Practice: Find the range for the data sets on the previous section for a through e.

28
Mixed Practice

a) Ammon makes \$6.45 per hour. He worked 5.3 hours Monday, 8 hours on Tuesday, 2.75 hours on
Wednesday, 0.75 hours on Thursday, and 8 hours on Friday. How much did he make that week?

b) Lacey has a piece of wire that is 3.5 feet long. She needs to cut 4 pieces off that are .85 feet
each. How much wire is left?

c) Last time I filled the tank, the mileage read 1,345.8. Now it reads 1383.2. How far did I drive on
the last tank of gas? If I use 15 gallons, what was the miles per gallon?

d) 3.2 pounds at 69 cents per pound of potatoes, 2.5 pounds of carrots at 40 cents per pound, and
0.75 pounds of onions at \$1.20 per pound are bought for stew. What is the total cost?

e) Women live, on average to 73.5 years old. Men live to 68.9 years old. How much longer do
women live on average?

f) Drive Safe car rental charges \$34 per day and \$0.04 per mile. How much would it cost to rent a car
for 3 days for a 300 mile round trip?

g) We took in \$350 at the garage sale last week. We spent \$14.40 on advertising and \$40 on meals
for the families working the sale. We will split the profits evenly among the six families.

h) Tad charges \$0.015 per square meter to maintain a lawn each week. What is the bill for a yard that
is the cost for a 30.2 m by 79.5 m lawn?

i) Diane bought a jacket for \$45.80, three scarves for \$0.78, and a pack of gum for 25 cents. she
multiplies by 0.06 to find tax. How much does she pay at the check out?

j) Larry walks on the tread mill daily. Last week he burned 325, 287, 295, 290, 340, 345, and 340
calories. What is the mean, median, mode and range of this data?

29

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