Multiplication of Decimals

 

  1. General Multiplication of Decimals

    The rules for multiplication of decimals come from the rules of multiplying fractions.  Consider the product

            0.02 x 0.013

    in fraction form this is

                2                13               26
                          x                =                   =  0.00026 
               100           1000          100000

    Notice that the denominator is found by adding the two zeros from the first denominator to the three zeros of the second denominator.  In decimal notation, we can see that there are 2 digits to the right of the first decimal point and three digits to the right of the second decimal point.  The result has 2 + 3 = 5 digits to the right of the decimal point.

    In general to multiply two decimal numbers, follow the steps below

    Step 1    Multiplying the numbers ignoring the decimal points.

    Step 2    Add the number of digits to the right of the decimal points.

    Step 3    Place the decimal point so that the number of digits to the right is the result of part 2.



    Example

            0.04 x 0.3


    Solution
    1. 4 x 3  =  12
    2. 2 + 1  =  3    0.04 has 2 digits to the right and 0.3 has one digit to the right
    3. 0.012        0.012 has 3 digits to the right of the decimal point

    Example

            1.02 x 0.016


    Solution

     

    1.      102
          x 16
           612
      + 1020
          1632
    2. 2 + 3 = 5
    3. 0.01632

     

    Example

            2.3 x 47


    Solution

     

    1.         2
                 1
             23
          x 47
           161
        + 920
          1081
    2. 1 + 0 = 1    A whole number has 0 decimal digits.
    3. 108.1

    Exercises
       
    Multiply the following decimals

    1. 0.32 x 0.0006        0.000192

    2. 8.13 x 39        317.07

     

  2. Multiplying a Decimal by a Power of 10

    Consider the multiplication problem

            0.00034 x 100

    In fractions this becomes

                      34                   1 100            34
                                        x                =                   =  0.034 
                   1000 100000              1             1000

    Notice that we canceled 2 zeros in the fraction.  This results in moving the decimal point over to the right two places.  

    In General

    When multiplying a decimal by a power of 10, move the decimal to the right the same number of places as the number of zeros of the power of 10.


    Example

    Multiply

    3.416 x 100


    Solution

    Since there are two zeros in 100, we move the decimal two places to the right

            3.416 x 100    =  341.6



    Example

    Multiply

            2.05 x 10,000    


    Solution

    Since there are four zeros in 10000, we move the decimal four places to the right.  In order to do this, pad 2.05 with 2 extra zeros first.

            2.05  =  2.0500

            2.0500 x 10,000    =  20,500


    Exercises

    Multiply
    1. 34.30541 x 1000    34,305.41

    2. 97.51 x 100,000    9,751,000



  3. Applications

    Example

    How many meters is 3.4 kilometers

    Solution

    We use the fact that there are 1000 meters in 1 kilometer.  This becomes a multiplication problem involving a power of ten

            3.4 x 1000  =  3.400 x 1000  =  3,400

    There are 2,100 meters in 2.1 kilometers.

    Example

    You just bought a 2.1 gigahertz Intel processor. How many hertz is this given that one gigahertz is 1,000,000,000 hertz.  We multiply, noting that this is a power of 10.

            2.1 x 1,000,000,000  =  2.100000000 x 1,000,000,000 

            =  2,100,000,000

    The processing power is 2,100,000,000 hertz (two billion one hundred million hertz).

    Example

    You have just bought 3.2 pounds of rib eye steak at $2.89 per pound.  What will the steak cost you?

    Solution

    This is the multiplication problem 2.89 x 3.2.


             289
          x   32
             578
        + 8670
           9248

    Adding decimal digits gives 

            2 + 1  =  3

    2.89 x 3.2  =  9.248

    Since this is a money problem, we should round to the nearest cent, which gives

            $9.25

    The steak will cost $9.25.


    Example

    You have just bought a building on highway 50 that has a rectangular dance floor of dimensions 10.3 feet by 11.2 feet.  What is the square area of the dance floor?

    Solution

    Recall that the area of a rectangle is the length times the width.  We multiply

                112
            x  103
                336
         + 11200
            11536

    Adding decimal digits gives

            1 + 1  =  2

    The floor is 115.36 square feet



    Exercises
    1. How many grams are there in 5.42 kilograms?    5,420 grams


    2. Your car achieves 24.5 miles per gallon on the highway and has a 14.2 gallon tank.  How many freeway miles can you drive on a full tank of gas?
       347.9 miles


    3. You just bought a house and your mortgage payments are $1004.23.  You will being making monthly payments for 30 years.  How much total money will you pay on this loan?    $361,522.80

 

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