Mathematic Diagnostic Placement Test Study Guide Mathematics Diagnostic Placement Test by lisashepherd

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									Mathematic Diagnostic/Placement Test


     Study Guide
Mathematics Diagnostic / Placement Test                                                                          Example

                                                Test Directions
      •   Write out all calculations on the test. If necessary, use the back of the page.
      •   No books, notes, or calculators may be used during this test.
      •   Always reduce and simplify.
      •   For word problems, write your answer as a complete sentence.
      •   After you have solved each problem, circle the correct answer on this test sheet AND
      •   Mark your answer on the separate answer sheet provided.


                                                                                                          Answers
1.        Round 69,456 to the nearest hundred.                                                            1. 69,460

2.        The attendance at four night baseball games was 6,380 the first night,
          5,963 the second night, 6,754 the third night, and 7,018 the last night.                        2. 26,115
          What was the total attendance for the four nights?

3.        Divide: 72 ÷ 0                                                                                  3. Undefined

4.        Adams’ grocery came to a total of $15.97. How much change he get from $20?                      4. $13.97

5.        Divide: 43, 800 ÷ 600                                                                           5. 73

6.        What is the product of 15 and 409?                                                              6. 6,135
                    4 5                                                                                        19
7.        Add:        +                                                                                   7. 1
                    5 6                                                                                        30
8.        Perform the indicated operations: 63 ÷ 7 + 5 ⋅ 2 - 6                                            8. 13
                   5 15                                                                                        2
9.        Divide:    ÷                                                                                    9.
                   8 16                                                                                        3
                                                              3
10.       Perform the indicated operations:       212 − 3(6 + 4 )                                          10. 2
                              1    3                                                                               9
11.       Subtract:         3 −2                                                                           11.
                              5    4                                                                            20
12.       Find the difference between 3.9 and 0.22. Round the difference to the nearest tenth.             12. 3.7

13.       Divide: 7.8 ÷ 0.6                                                                                13. 13
                                            3
14.       Multiply:          73, 485 × 10                                                                  14. 73,485,000

15.       What is the place value for the digit 7 in 893,527,000?                                          15. Thousand

16.       Find the quotient of 6027 and 49.                                                                16. 123
                     2     1                                                                                     2
17.       Divide: 3 ÷ 2 .                                                                                  17. 1
                     3     5                                                                                     3
                                                                           2 6
18.       Is the following statement true or false? Explain your answer.    <                              18. False
                                                                           3 7
19.       Write this percent as a decimal: 7%                                                              19. 0.07

20.       Change to a percent: 17/20                                                                       20. 85%
                                                                                                                 3
21.       Change to a fraction: 0.075                                                                      21.
                                                                                                                40
22.       Find the average of 1020, 911, 1145, 987, and 1005.                                              22. 1267
                              3 4                                                                               1
23.       Multiply:            ×                                                                           23.
                              8 9                                                                               6
24.       List the set of decimals from smallest to largest: 0.5009, 0.509, 0.50099, 0.50909               24. 0.5009, 0.50099,
                                                                                                               0.509, 0.50909
25.       Find the area of a square that is 0.12m on a side. (Round the area to the nearest hundredth.)    25. 0.01m 2
Whole Numbers
The whole numbers are 0, 1, 2, 3, 4, 5, and so on. All whole numbers can be written using ten digits. Numbers larger than 9
are written in place value form by writing the digits in positions having standard place value.


Hundred     Ten                     Hundred     Ten                   Hundred      Ten
Billions    Billions     Billions   Millions    Millions   Millions   Thousands    Thousands   Thousands   Hundreds        Tens   Ones


           Billions                            Millions                           Thousands                    Ones (unit)
Rounding Whole Numbers
       To round a whole number means to give an approximate value.
Example 1:    Round 827,456 to the nearest ten thousand.
              827,456 Underline the ten thousands place. The digit to the right of the underlined digit is 7, which is
              greater than 5. Therefore, add 1 to the digit above the line. (2 + 1 = 3.)
              830,000 Replace all digits to the right of the underlined digit with zeros.
              Therefore, 827,456 ≈ 830,000
              .
Practice Problems                                                                                                     Answers
     1.    What is the place value for the digit ‘4’ in 1,234,567?                                                    1. thousands
     2.    Round 74,572 to the nearest hundred.                                                                       2. 74,600

Comparing Whole Numbers
The symbols equal (=), not equal ( ≠ ), less than (<), greater than (>), less than or equal ( ≤ ), and greater than or equal
( ≥ ) are used to compare two whole numbers.
Example 2:         Insert > or < between the numbers to make a true statement.
          a)       74 ___76                           b)       1,260 ___ 1,206
                   74 < 76                                     1,260 > 1,206

Adding Whole Numbers
The result of adding is called the sum (or total).
Example 3:        Add: 45,329 + 30,397
                           11
                     45, 3 2 9          Align each like place value.
                   + 30, 397            Add and regroup.
                    75,726              9 + 7 = 16 = 1 ten + 6 ones
                                        1 + 2 + 9 = 12 tens = 1 hundred +2 tens
                       45,329 + 30,397 = 75,726

Subtracting Whole Numbers
The result of subtracting is called the difference.
Example 4:         Subtract. 3,032 – 576
              9 12
           2 10 2 12
               / /
           3, 0 3 2 To subtract in the ones column, borrow 1 ten from the tens column and add it to the ones column to get
           / / / /
         – 5 7 6 12 in the ones column; then subtract: 12 – 6 = 6.
          2, 4 5 6 To subtract in the tens column, borrow 1 thousand from the thousands column and write it as 10 hundreds
in the hundreds column; borrow 1 hundred from the hundreds column and think of it as 10 tens; add these 10 tens to the 2
tens that is left in the tens column to get 12 tens; then subtract: 12 – 7 = 5;
Subtract in the hundreds column: 9 – 5 = 4; Subtract in the thousands column: 2 – 0 = 2. The difference is 2,456.


Practice Problems                                                                                                     Answers
     1.    Compare 7,470 7,850 using < , > or = .                                                                     1.   7,470 < 7,850
     2.    976 + 427 + 38                                                                                             2.   1,441
     3.    Find the sum of 76; 3,418; 51 and 9. Round your answer to the nearest ‘tens’ place.                        3.   3550
     4.    Find the difference: 746 – 378                                                                             4.   368
     5.    Subtract 759 from 3,800.                                                                                   5.   3,041
Multiplying Whole Numbers
The product is the answer to a multiplication exercise.
        To multiply by a one-digit number:
Multiply each digit in the top number by the one-digit number (beginning with the ones place.)
        To multiply by a two- or greater-digit number:
Multiply each digit in the top number by each digit in the bottom number (beginning at the ones place); add the products.
Example 5:                  159 × 48
                  159
                 × 48                Regroup:           48 = 40 + 8
                1,272                159 × 8        Begin multiplying with the ones digits. If the product is ten or more, carry
             + 6,360                 159 × 40        the tens digit to the next column and add it to the product in that column.
               7,632                                to the product in that column.
                                                    Repeat the process for every digit in the second factor.
                                                    When the multiplication is complete, add to find the product.

Practice Problems                                                                                                Answers
    1.   74 × 58                                                                                                 1. 4,292
    2.   Find the product of 572 and 121.                                                                        2. 69,212

Dividing Whole Numbers
The quotient is the answer to a division exercise.
Example 6:        28 ÷ 2 = 14                                  The answer to a division problem can be checked by multiplication.
                                                                               14
                   28
                        = 14             or      28 ÷ 2 = 14         or      2 28                  Check:            14 × 2 = 28
                  2
When a division exercise does not come out evenly, the quotient is not a whole number.
Example 7:       In     57 ÷ 9 = 6 R3 ,                                 The quotient is written 6 R 3.

                               Note: Division by zero is not defined. It is an operation that cannot be performed.

Dividing Whole Numbers
The acronym DMSB tells you in what order to do long division.
         To divide whole numbers, follow these steps:
Step 1: Divide, writing a digit in the answer.
Step 2: Multiply the digit you placed in the answer by the divisor.
Step 3: Subtract the product you obtained from Step 2 from the subtrahend above.
Step 4: Bring Down the next digit in the dividend to be divided.
Example 7:                 Example 8:                Example 92:             Example 10:
                                                                   27                   11R1
              6                         6 R1                   2 54                  21 232
           6 36                       6 37                        −4                   − 21
            − 36                        − 36                      14                      22
               0                           1                    − 14                    − 21
                                                                    0                     1

Practice Problems                                                                                                Answers
    1.   287 ÷ 7                                                                                                 1. 41
    2.   8,436 ÷ 23                                                                                              2. 366R18
    3.   Find the quotient of 840,000 and 700.                                                                   3. 1,200

Whole Number Exponents
Whole number exponents greater than 1 are used to write repeated multiplication in shorter form.
Example 11:    25 = 2 ⋅4 ⋅ 2 ⋅43 2 = 32
                    1 22 2⋅                                           Example 12:        121 = 12
                               5 times


Practice Problems                                                                                                Answers
    1.   62                                                                                                      1. 36
                                                                                                                      6
    2.   Write in exponential form: 4 × 4 × 4 × 4 × 4 × 4                                                        2. 4
Order of Operations
         To evaluate an expression with more than one operation, perform each operation in the following order:
P   = Parentheses or any grouping symbols – Do the operations within grouping symbols first.
E   = Exponents – Do the operations indicated by exponents.
M   = Multiplication or
D   = Division – Do only multiplication and division as they appear from left to right.
A   = Addition or
S   = Subtraction – Do addition and subtraction as they appear from left to right.

                                                Note: MD = Multiplication and Division – done in order from left to right.
                                                      AS = Addition and Subtraction – done in order from left to right.

The acronym PEMDAS helps you remember the Order of Operations.
Example 13:   Perform the indicated operations.
              (2 +3) × 5 = 5 × 5 = 25 Do the operation inside parentheses first.
Example 14:   Perform the indicated operations.
                   12 − 8   ÷ 2(7 – 5) + 3 2 × 4 = 12 − 8 ÷ 2(2) + 9 × 4 = 12 − 4(2) + 9 × 4 = 12 − 8 + 36 = 4 + 36 = 40
Practice Problems                                                                                             Answers
     1.   9 ÷ 3 + 4× 5 – 2                                                                                    1. 21
     2.    (33 – 80 ÷ 4) 2 + 120                                                                              2. 626

Finding the Average (Mean) of a set of numbers
         To find the average (mean):
Find the sum of the numbers (add the numbers) first; then divide the sum by the number of items.
Example 15: Find the average 78, 99, 80, 83
               First find the sum of the numbers: 78 + 99 + 80 + 83 = 340
               Then divide the sum by the number of items: 340 ÷ 4 = 85
               Therefore, the average (mean) of the test scores is 85.

Practice Problems                                                                                             Answers
     1.   Find the average of 57, 34, 65                                                                      1. 52
     2.   If a class of 14 people got the scores 78, 74, 78, 74, 78, 78, 78, 78, 99, 80, 83,                  2. 81
          80, 92, 84 for a Math Final Exam, find the mean (average) of the test scores.

Fractions and Mixed Numbers
             a
A fraction     is a name for a number that represents part of a whole quantity. A fraction is a part of a whole unit.
             b
                                                Note: Since division by zero is undefined, the denominator can never be zero.

A proper fraction is one in which the numerator is less than the denominator.
An improper fraction is one in which the numerator is greater than or equal to the denominator.
A mixed number is the sum of a whole number and a fraction (proper) with the plus sign left out.
                  3         2                        3             3
Example 16:     1 and 5 are mixed numbers. 1 means 1 + where the plus sign is not written.
                  5         7                        5             5

Simplifying Fractions
Equivalent (equal) fractions are fractions that are different names for the same number.
Simplifying (reducing) a fraction is the process of renaming it resulting in a smaller numerator and denominator.
A fraction is completely simplified when its numerator and denominator have no common factors other than 1; it is then
called a fraction in lowest terms.
To find an equivalent fraction, divide (reduce) or multiply (build) the numerator and denominator by the same number
(common factor.)
                   12 12 ÷ 2 6 ÷ 3 2                Reduced fraction          12 12 ⋅ 2 24            Built-up fraction
Example 17:            =        =       =            in lowest terms              =      =             in higher terms
                   18 18 ÷ 2 9 ÷ 3 3                                          18 18 ⋅ 2 36


The process of dividing out the common factors from the numerator and denominator is called canceling.
          To simplify (reduce) a fraction completely:
Eliminate (divide out, also called canceling) all common factors other than 1 in the numerator and in the denominator.
Divide both numerator and denominator by any common factor. Continue dividing by common factors until no common
factors remain.
                            18                 18 18 ÷ 3     6÷3 2              18 18 ÷ 9 2
Example 18:       Simplify      .                  =      =        =      or        =       =
                            45                 45 45 ÷ 3 15 ÷ 3 5               45 45 ÷ 9 5

Practice Problems                                                                                                        Answers
                                                                   12                                                         3
    1.   Simplify completely (reduce to the lowest term.)                                                                1.
                                                                   16                                                         4
                                                                    96                                                        2
    2.   Simplify completely (reduce to the lowest term.)                                                                2.
                                                                   144                                                        3

Writing Fractions and Mixed Numbers
          To change an improper fraction to a mixed number:
First, simplify the fraction, if possible. (This makes the process simpler.)
Then divide the numerator by the denominator; the quotient will become the whole-number portion of the mixed number;
the remainder will become the numerator of the fractional portion of the mixed number; the divisor will become the
denominator of the fractional portion of the mixed number. Finally, reduce to lowest terms, if possible.

                                25                                                                            202
Example 19:           Write                as a mixed number.                    Example 20:        Write           as a mixed number.
                                   4                                                                          22
                       1                                                                                                              2
                  6                                                                                                           9
                    4                                                                                                         11
                                       25                      1                              202       101
                 4 25                       = 25 ÷ 4 = 6                         Reduce:            =         Divide:     11 101
                                       4                       4                               22       11
                      24                                                                                                          99
                      1                                                                                                           2
                                            25        1                                       202        2
                      Therefore,                 =6                              Therefore,         =9
                                            4         4                                       22         11

         To change a mixed number to an improper fraction:
Multiply the whole number by the denominator;
add the result to the numerator;
place the result as a new numerator and keep the given denominator.
                           1
Example 21:         Write 3 as an improper fraction.
                           5
                      1 3 ⋅ 5 + 1 16
                    3 =          =
                      5     5      5

Practice Problems                                                                                                        Answers
                 38                                                                                                               1
    1.   Write         as a mixed number.                                                                                1. 6
                  6                                                                                                            3
                      2                                                                                                       65
    2.   Write 21          as an improper fraction.                                                                      2.
                      3                                                                                                       3

Multiplication of Fractions
        To multiply fractions:
Simplify (if possible); write the product of the numerators over the product of the denominators.
To simplify the process, there is a shortcut for multiplying fractions called cancellation.
                               3
                      3        15           3×3           9
Example 22:                ×           =              =                  Be sure every answer is reduced to lowest terms.
                      5        19          1 × 19         19
                       1
Multiplying Mixed Numbers
          To multiply mixed numbers:
First write each mixed number as an improper fraction;
Simplify (if possible); write the product of the numerators over the product of the denominators.
                       3           1           11          5       55           7
Example 23:        2       ×2          =               ×       =           =6
                       4           2           4           2       8            8

Practice Problems                                                                                                                          Answers
                                       6 7 4                                                                                                    3
    1.   Multiply.                      × ×                                                                                                1.
                                       7 8 5                                                                                                    5
                                           1           3                                                                                          3
    2.   Multiply.                  2          ×1                                                                                          2. 3
                                           4           5                                                                                          5

Division of Fractions
If two fractions have a product of 1, each fraction is called the reciprocal of the other.
                    3                     5             3 5
Example 24:           is the reciprocal of , because × = 1 .
                    5                     3             5 3
Zero (0) does not have a reciprocal.


Practice Problems                                                                                                                          Answers
                                                           23                                                                                   24
    1.   Find the reciprocal.                                                                                                              1.
                                                           24                                                                                   23
                                                               3                                                                                 4
    2.   Find the reciprocal.                              2                                                                               2.
                                                               4                                                                                  11

             To divide fractions:
Multiply the first fraction by the reciprocal of the divisor (the second fraction); that is, invert and multiply.
For division, you multiply the first fraction by the reciprocal of the second fraction.
                                                                                1
                     3         1               3       12              3        2⋅ 2 ⋅ 3          18        4
Example 25:                ÷           =           ×           =            ⋅                =         =2
                     14        12          14              1       2⋅ 7              1            7         7
                                                                   1


                                                                                         Note: Change an improper fraction answer to a mixed number.

Dividing Mixed Numbers
          To divide mixed numbers:
First write each mixed number as an improper fraction; Then proceed as with fraction.


         Note: When divide fractions, multiply the first fraction by the reciprocal of the second fraction.

                                                                        5
                       3           1           15          21          15           5        25
Example 26:        3         ÷4         =              ÷           =        ×            =
                       4           5               4           5        4           21       28
                                                                                    7

Practice Problems                                                                                                                          Answers
                     9         12                                                                                                                 1
    1.   Divide.           ÷                                                                                                               1. 1
                     10        15                                                                                                                 8
                         1          5                                                                                                             1
    2.   Divide. 2             ÷1                                                                                                          2. 1
                       16           6                                                                                                             8
Comparing Fractions and Mixed Numbers
         To compare fractions:
Rewrite (reduce or build, if necessary) fractions with a common denominator; compare numerators.
A common denominator is a number that can be divided evenly by all denominators in a problem.
The smallest number that can be divided evenly by all denominators in a problem is called the least common denominator
or LCD.

                                     Note: Sometimes the largest denominator in the problem is the least common denominator.

                                     3               1
Example 27:     Compare           .          and
                         8      5
                  The LCD of 8 and 5 is 40. Build each fraction so it has 40 as a denominator.
                   3           3⋅5            15                         1       1⋅ 8       8
                           =             =               and                 =          =                        Compare the numerators: 15 > 8.
                   8           8⋅5            40                         5       5⋅8        40
                                                                 3       1
                  Compare the fractions:                             >
                                         8 5
         To compare mixed numbers:
First compare the whole-numbers; if the whole numbers are the same, compare the fractional portions.
                                         5               2                                                                  5           4
Example 28:     Compare 1                     and 3          .                      Example 29:                 Compare 2       and 2       .
                        6         3                                                                          8        7
Compare the whole numbers: 1 < 3.                                                   Compare the whole numbers: 2 = 2.
                       5             2                                                                                                                      5         4
                   1           < 3                                                Since the whole numbers are the same, compare the fractions:                  and       .
                       6             3                                                                                                                      8         7

      To list fractions from smallest to largest (or from largest to smallest, according to directions):
Build the fractions (if needed) so that they have a common denominator;
list the fractions (with common denominators) with numerators from smallest to largest (or from largest to smallest);
simplify.
                                                    2 3         3
Example 30:         List from smallest to largest: , and .
                                                    3 8         4
                                                                                              2 16       3   9     3 18
                    The LCD of 3, 8, and 4 is 24. Build the fractions to a denominator of 24.   =          =          =
                                                                                              3 24 8 24 4 24
                                                                       9 16 18
                    List the fractions in the order of the numerators:    ,   ,
                                                                       24 24 24
                                                                3 2       3
                    Refer to the original fractions, the list is , , and .
                                                                8 3       4


Practice Problems                                                                                                                                  Answers
                       4             6                                                                                                                  4       6
    1.   Compare            and          .                                                                                                         1.       >
                       3             7                                                                                                                  3       7
                           3       16                                                                                                                    3     16
    2.   Compare 5           and 5                                                                                                                 2. 5    < 5
                           8       36                                                                                                                    8     36
                                                                                  7 2 3 5                                                               2 3 7 5
    3.   List the fractions from smallest to largest                               , , ,                                                           1.     , , ,
                                                                                  9 3 4 6                                                               3 4 9 6


Adding Fractions
         To add fractions:
Build each fraction to a common denominator (LCD); add the numerators and write the sum over the common denominator;
simplify, if possible.
                                     1           3       1       3       1       3⋅ 2       1       6       7
Example 31:       Add.                       +       =       +       =       +          =       +       =
                                     8           4       8       4       8       4⋅2        8       8       8
Adding Mixed Numbers
        To add mixed numbers:
Add the fractions and the whole numbers separately; simplify the fractional sum, if possible.
                                             1            3
Example 32:         Add                  7       +5           .
                                             2            4
                        1                    1                2
                    7       =7+                  =7+                                       The LCD of 2 and 4 is 4. Build the fractions. Add.
                        2                    2                4
                        3                3                    3
                +5          =5+                  =5+
                        4                4                    4
                                                              5                    1           1
                                                  12 +            = 12 + 1             =13
                                                              When a sum includes a whole number and an improper
                                      4         4      4
fraction, the improper fraction must be converted to a mixed number and then added to the whole number.


Practice Problems                                                                                                                                                    Answers
                1       7                                                                                                                                                     3
    1.   Add        +       .                                                                                                                                        1. 1
                2       8                                                                                                                                                     8
                 1          7                                                                                                                                                 3
    2.   Add 1       +           .                                                                                                                                   2. 1
                 6          12                                                                                                                                                4

Subtracting Fractions
         To subtract fractions:
Build each fraction to a common denominator (LCD); subtract the numerators and write the difference over the common
denominator; simplify, if possible.
                                                              2       3       2⋅5          3⋅3          10       9        1
Example 33:         Subtract                                      −       =            −            =        −        =
                                                              3       5       3⋅5          5⋅3          15       15       15
Subtracting Mixed Numbers
         To subtract mixed numbers:
Subtract the fractions and the whole numbers separately; simplify the fractional difference, if possible.

 Note: When the top fractional portion is smaller than the bottom fractional portion, we must “rename” the top mixed
       number. Borrow “one” from the top whole number. Convert the borrowed “one” to a fraction with the same
       denominator as the existing fraction.
                                             2            4
Example 34:         Subtract 5                   −3           .               The LCD of 3 and 5 is 15. Build the fractions.
                                             3            5
                                                                                                                               15
                                                                                                             1 converted to

                                                                      borrowed from 5
                                                                                                                      }        15

                            2                    10                          }                 10                     15                10         25        25
                        5        =5+                  =4+                      1           +        =4+                             +        =4+        =4
                            3                    15                                            15                     15                15         15        15
                            4                    12                                                                                                           12
                     −2          =2+                  =                                                                                                  −2
                            5                    15                                                                                                           15
                                                                                                                                                             13
                                                                                                                                                         2         Subtract.
                                                                                                                                                             15

Practice Problems                                                                                                                                                    Answers
                        1        3                                                                                                                                        1
    1.   Subtract           −        .                                                                                                                               1.
                   2 8                                                                                                                                                    8
                    1  2                                                                                                                                                       7
    2.   Subtract 7 − 4 .                                                                                                                                            2. 2
                    4  3                                                                                                                                                      12
Decimals
Decimal numbers, or numbers written in decimal notation, more commonly referred to as decimals, are another way of
writing fractions and mixed numbers. A decimal represents a part of a whole.
Place value — the value of a digit depends on its placement.
Example: The place values of the decimal 9,253.481 are shown in the figure below:

                          Thousands                                  Thousandths
                               Hundreds                           Hundredths
                                    Tens                     Tenths
                                               Ones

                             9
                             1       5
                                 422 4 3 3            .     4
                                                            14   28 4 1
                                                                    3
                              whole number places            decimal places
                                                decimal point
Rounding Decimals
Exact decimals are decimals that show exact values.
Approximate decimals are rounded values.
           To round a decimal to a given place value:
Determine the place value to which you will round (by underlining the given place value), and then examine the digit to the
right. If the digit to the right is 5 or greater, add one to the digit above the line (rounding up); if the digit to the right is less
than 5, do not change the digit above the line (rounding down.) Replace all digits to the right of the underlined digit with
zeros or leave blank.
Example 35:          Round 8.27 to the nearest tenth.
                     8.27 Underline the tenths place. The digit to the right of the underlined digit is 7, which is greater than 5.
                             Add 1 to the digit above the line. Therefore, 8.27 ≈ 8.3

Changing Decimals to Fractions
         To change a decimal to a fraction:
Write the digits of the decimal places (starting with the first non-zero digit) as the numerator of the fraction;
count the number of decimal places which determines the number of zeros in the denominator;
write a denominator of “1” followed by the number of zeros determined.
Example 36:        Change 0.38 to a fraction and simplify, if possible.
         Write 39 as the numerator. Since 0.39 has two decimal places, insert two zeros following the “1” in the denominator.
                                               19
                                    3838   19
                   0. 38 =
                       {           =     =                    Reduce.
                              1 00   100   50
                     2 places   {               50
                                   2 zeros
Comparing Decimals
        To compare decimals:
Align each place value beginning with the largest place value, working from left to right;
compare digits within a column; use >, <, or =.
Example 37:      Compare 2.14 and 2.13.
                               2. 1 4
                               2. 1 3

                      Same 2 = 2
                      Same 1 = 1
                      Different 4 > 3
                  Therefore, 2.14 > 2.13.
To ease the comparison, extra zeros can be written to the right of the decimal point, if necessary, so that the number of
decimal places is the same.

Practice Problems                                                                                        Answers
    1.   Round 2.8453 to the nearest hundredth.                                                          1. 2.85
                                                                                                              4
    2.   Change 0.16 to a fraction and simplify, if possible.                                            2.
                                                                                                             25
    3.   List 11.5, 11.2, and 11.25 in order from smallest to largest.                                   3. 11.2, 11.25, 11.5
Adding Decimals
          To add decimals:
Write the decimals in columns with decimal points aligned (inserting extra zeros to align place values);
add the decimals as though they are whole numbers;
align the decimal point in the sum with those above.

                                                    Note: Any whole number is understood to have a decimal point at its right.

Example 38:       Add: 2.54 + 0.008 + 11.
                  Write each number with three decimal places and align place values and decimal points:
                  2.540                             The extra zeros help line up place values.
                  0.008
               + 11.000
                 13.548            Add.             2.54 + 0.008 + 11 = 13.548

Subtracting Decimals
          To subtract decimals:
Write the decimals in columns with decimal points aligned (inserting extra zeros to align place values);
subtract the decimals as though they are whole numbers;
align the decimal point in the difference with those above.

Example 39:       Subtract: 37.42 – 21.57
                     6 1312
                  37.4 2           Place the larger number on top and line up place values and decimal points.
                 –21.57            Subtract by borrowing.
                  15.85            Bring down the decimal point.
                  37.42 – 21.57 = 15.85

Practice Problems                                                                                              Answers
    1.   Add 10.403 + 0.75                                                                                     1.   11.153
    2.   Add 9.06 + 0.82 + 11.5 + 4.35. Round to the nearest tenth.                                            2.   25.7
    3.   Subtract. 14.6 – 0.475.                                                                               3.   14.125
    4.   Subtract 0.005 from 0.55. Round to the nearest hundredth.                                             4.   0.55


Multiplying Decimals
          To multiply decimals:
Multiply the numbers as though they are whole numbers;
the total number of decimal places in the factors is the number of decimal places the product must have;
if necessary, insert zeros to the left of the product so that there are the number of decimal places needed.


                                                      Note: There is no need to align place values when multiplying decimals.

Example 40:       Multiply: 1.07      × 3.5.                     Example 41:       Multiply: 0.49   × 0.05.
                     1.07                2 places                                  0.49               2 places
                   × 3.5                + 1 place                                × 0.05               2 places
                     535                                                         0.0245               4 places
                 + 321                                                            Insert extra zeros as placeholders.
                   3.745             3 places
                  1.07 × 3.5 = 3.745                                               0.49   × 0.05 = 0.0245

Practice Problems                                                                                              Answers
    1.   Multiply 1.02   × 2.34, round the product to the nearest thousandth.                                  1. 2.387

    2.   Multiply 0.832     × 6.01.                                                                            2. 5.00032
Multiplying Decimals by Powers of 10
A power of 10 is the value obtained when 10 is multiplied/divided one or more times by 10 or written with an exponent.
                                                                                        5
Example 42:                                                1 44 10 4 43
                  100,000 is a power of 10 since 100,000 = 10×10×2 ×10×10 = 10
                                                                         5 times
The exponents may be used to represent the powers of 10.
                                                                              3
Example 43:                                            1 104
                  1,000 is a power of 10 since 1,000 = 10×2 ×10 = 10
                                                        4 3
                                                               3 times
Multiplying by powers of 10 can be accomplished by moving decimal point:

        To multiply any decimal by a power of 10, such as 10, 100, 1,000 and so on:
Count the number of zeros in the of the power of 10; move the decimal point in the decimal that many places to the right.

                                             Note: You may need to add zeros to the right or left to accomplish the operations.

Example 44:       Multiply. 0.65 × 1,000.
                  0.65 × 1,000 = 0.6 5 . × 1, 000 = 650. = 650
                                              {
                                                     3 zeros
                                    3 places    Move the decimal point three decimal places to the right, adding one zero.
         To multiply by positive powers of 10 in exponential form:
Look at the exponent; move the decimal point that many places to the right.
                                         2
Example 45:       Multiply. 0.39 × 10 .
                            2                    2
                  0.39 × 10 = 0 . 3 9 . × 10 = 39. = 39              Move the decimal point two decimal places to the right.

                                2 places
Practice Problems                                                                                                Answers
    1.   Multiply 4.93 × 10,000.                                                                                 1. 49,300
                                3
    2.   Multiply (3.8212)( 10 ).                                                                                2. 3,821.2

Dividing Decimals
          To divide decimals by decimals:
If the divisor is not a whole number, move the decimal point in both the divisor and dividend to the right the number of
places necessary to make the divisor a whole number; (Note: Always start with divisor!)
Place the decimal point in the quotient above the decimal point in the dividend;
Divide as if both numbers were whole numbers;
Round to the given place value. (If no round-off place is given, divide until the remainder is zero or round as appropriate in
the problem.
Example 46:         3.456 ÷ 0.02
Change the divisor into a whole number by multiplying it by powers of ten.
Do this by shifting decimal points. 0 .0 2 3.456
                                       ∪∪
If you multiply the divisor by a power of ten, you must multiply the dividend by the same power of ten. 0 .02 3.456
                                                                                                          ∪∪ ∪∪
                                                                                                           172.8
Then proceed as you did above (Bring the decimal point up above its new position and divide.             2 345.6
                  3.456 ÷ 0.02 = 172.8

                  Note: When told to divide, look for a terminating or repeating decimal.
                       When told to round, divide until one place past the place required in your answer, then round.


Practice Problems                                                                                                Answers
    1.   Divide 0.6 ÷ 0.2 .                                                                                      1. 3
    2.   Divide 0.4 6.29 and round to the nearest hundredth.                                                     2. 15.73
Changing Fractions to Decimals
         To change a fraction to a decimal: Divide the numerator by the denominator.
Or, when appropriate, follow this rule:
         To change a fraction to a decimal:
Build a fraction with a denominator that is a power of 10; change the fraction into a decimal.
                        3
Example 47: Change to a decimal.
                        5
Build the fraction or fractional portion, if possible.         Divide numerator by denominator.
3 3× 2   6                                                      3
 =     =   = 0.6                                                    = 3 ÷ 5 = 0.6       Terminating decimal; no need to round.
5 5 × 2 10                                                      5

Converting Decimals to Percents
Multiply the decimal by 100 (move the decimal point two places to the right) and put the % symbol.

Example 48:       0.75 = 0.75. = 75%

Example 49:       1.02 = 1.02. = 102%

Converting Percents to Decimals
Divide the decimal by 100 (move the decimal point two places to the left) and remove the % symbol.

Example 50:       80% =0 .80. = 0.8

Example 51:       120% = 1.20. = 1.2


Practice Problems                                                                                             Answers
                  3
    1.   Change to a decimal.                                                                                 1. 0.75
                  4
                 12
    2.   Change      to a decimal and round to the nearest tenth.                                             2. 0.5
                  23
    3.   Convert 0.654 to a percent.                                                                          3. 65.4%
    4.   Write 0.4% as a decimal.                                                                             4. 0.004

Geometry
Perimeter
Perimeter is the measurement of the distance around a flat figure.
To find the perimeter of common geometrical figures such as square, rectangle, parallelogram, circle, use the appropriate
formulas.
Square                       Rectangle                             Parallelogram


                                       w                                            a


          s                                     l                                          b
      P = 4s                                 P = 2l + 2w                             P = 2a + 2b
The circumference of a circle is the distance around the circle.
Circle                                                To find the circumference of a circle
                                                      If C is the circumference, d is the diameter, and r is the radius
                                                      of a circle, then the circumference is the product of π and the diameter,
               Diameter                               or the product of π and twice the radius.
                                             Radius             C= πd             or           C = 2π



            Center
                                           Circumference
Area
Area is a measure of surface that is the amount of space inside a two-dimensional figure.
To find the area of common geometrical figures such as triangle, square, rectangle, parallelogram, trapezoid, circle, use the
appropriate formulas.

Square                                Rectangle                                    Parallelogram


                                           w
                     s                                          w                                  h

         s                                              l                                                  b
                 2
     A=s                                            A=lw                                           A=bh

Triangle                                            Trapezoid                             Circle
                                                                     b1
                                                                                                       r
             h                                              h

                         b                                          b2
                     1                                          1
             A=        bh                               A=        ( b 1 + b 2 )h              A=   πr2
                     2                                          2

You may need to add zeros to the right of the dividend and continue to divide to see if the decimal repeats or terminates.


                         Note: To find the area of a polygon that is combination of two or more common figures, first divide it into
                               the common figure components.


Practice Problems                                                                                                  Answers
    1.   Find the perimeter of a rectangle with a length of 16 feet and a width of 4 feet.                         1. 40ft

    2.   Find the perimeter of a square whose sides measure 8 meters.                                              2. 32m

    3.   Find the circumference of a circle whose radius measures 2 yards.                .                        3.    ≈ 12.56yd
    4.   Find the area of a rectangle with a length of 16 feet and a width of 4 feet.                              4. 64ft 2
    5.   Find the area of a square whose sides measure 6 meters.                                                   5. 36m 2
    6.   Find the area of a circle whose radius measures 2 yards.                                                   6.   ≈ 12.56yd 2

								
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