ALGEBRA - solved problems
  Adding and subtracting decimals
48.    Add and subtract given decimals,
Solutions:   a)   7.9 + 12 + 0.147 = 20.047,           b)   215.04 + 3.756 + 17.3 = 236.096
                    7.900                                                215.040
                  12.000                                                    3.756
              +    0.147                                             +   17.300
                  20.047                                                 236.096
  c)   41.56 9.178 = 32.382,                d)   5.08 0.9937 = 4.0863
                  41.560                                                    5.0800
                  9.178                                                  0.9937
                  32.382                                                    4.0863
To add or subtract decimals, insert zeros in empty decimal place values so that all of the numbers have the same number of decimal places.
Write the corresponding decimal places one under another, i.e., the decimal points should be vertically aligned, and then follow the rules for adding or subtracting whole numbers. 
Line up the decimal point in the answer.
Multiplying and dividing decimals by powers of ten or by decimal units
49.    Multiply and divide given decimals by powers of ten.
Solutions:   a)   0.58 10 = 5.8,             b)   79.4 100 = 7940,            c)    0.001 10000 = 10,
  d)   30.07 10 = 3.007,      e)    0.08 100 = 0.0008,        f)   12.0 1000 = 0.012.
To multiply decimals by powers of 10, such as 10, 100, 1000 etc. move the decimal point right as many places as there are zeros in the power.
To divide decimals by powers of 10, move the decimal point left as many places as there are zeros in the power.
50.    Multiply and divide given decimals by decimal units.
Solutions:   a)   83.05 0.1 = 8.305,       b)   547.2 0.001 = 0.5472,       c)    0.9 0.0001 = 0.00009,
  d)    0.75 0.001 = 750,       e)    4.035 0.01 = 403.5,         f)   29.0 0.1 = 290.
To multiply decimal by the decimal units;  0.1, 0.01, 0.001, ... Move the decimal point of the decimal left as many places as there are zeros in the decimal unit.
Divide decimal by the decimal units by moving the decimal point of the decimal right as many places as there are zeros in the decimal unit.
Multiplication and division of decimal numbers
51.    Multiply given decimals.
Solutions:   a)   0.589 47.8,           b)     79.6 0.00503,           c)     629.7 1.03
      2356                             3980                                   6297
        4123                               000                                   0000
           4712                             2388                                  18891
      28.1542                      0.400388                                648.591
To multiply decimal numbers, ignore the decimal points and multiply the digits. Count the total number of decimal places in both decimal numbers being multiplied.
Place a decimal point in the answer so that it has as many digits to the right of the decimal point as the total number of decimal places in the two decimals being multiplied.
52.    Divide given decimal numbers.
Solutions:   a)     467.5 17 = 27.5,           b)     2.773 0.47 = (2.773 100) (0.47 100) =
     34                                                             = 277.3 47 = 5.9
        127                                                           235   
     119                                                                423               
            85                                                           423  
         85                                                                  0
              0
  c)   0.15059 3.7 = 1.5059 37 = 0.0407,       d)   10.8 0.004 = 10800 4 = 2700
                              150                                                           
                           148                                                               28
                                  259                                                        28 
                               259                                                             0
                                      0
To divide decimal numbers, first make the divisor into a whole number by multiplying both the dividend and the divisor by the same power of 10 (such as 10, 100, 1000, . . .) or move a decimal point in the divisor to the right end, and in the dividend the same number of places.
To divide a decimal by a whole number, divide as you would for whole numbers. Place a decimal point in the answer so that it lines up with the decimal point in the dividend.
If after dividing you have a remainder, add a zero to the dividend and continue to divide until there is no remainder or the decimals repeat.
  Rational numbers
53.    Compare given rational numbers.
Solutions:   since (-4) (-10) = 5 8 since 3 35 = (-7) (-15)
    since 5 7 = 8 4 since (-9) 3 < 5 (-5)
Equivalent rational numbers or fractions
54.    Write equivalent rational numbers or fractions.
Solutions:
Convert a rational number or fraction to an equivalent fraction by multiplying the numerator and denominator by the same nonzero number. Equivalent fractions are different fractions that represent the same number.
Reducing or simplifying rational numbers or fractions
55.    Reduce or simplify given rational numbers or fractions.
Solutions:
To reduce or simplify a fraction to lowest terms, divide the numerator and denominator by their greatest common divisor (gcd) or (greatest common factor - gcf).
Adding and subtracting rational numbers or fractions
56.    Add and subtract rational numbers or fractions.
Solutions:  
             12 = 2 2 3,      36 = 2 2 3 3,      48 = 2 2 2 2 3 
                      LCD(12, 36, 48) = 2 2 2 2 3 3 = 144
   
Find the least common denominator, write equivalent fractions, then add or subtract the fractions. Reduce if necessary.
Multiplication of rational numbers
57.    Multiply rational numbers or fractions.
Solutions:
Rational numbers are multiplied by multiplying numerators and multiplying denominators. Change any mixed numbers to improper fractions. Reduce fractions before multiplication.
Division of rational numbers, reciprocal fractions (multiplicative inverse)
58.    Divide rational numbers or fractions.
Solutions:
 
Dividing a rational number or fraction by another fraction is equivalent to multiplying the dividend with the reciprocal of the divisor.
Simplifying complex or compound fractions
59.    Simplify rational numbers or fractions.
Solutions:
A complex or compound fraction is a fraction whose numerator and/or denominator are also a fraction or mixed number. Divide the numerator by the denominator.
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