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Math Help - Leading Digit Estimate

  1. #1
    Member Sazza's Avatar
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    Arrow Leading Digit Estimate

    Okay i'm studying about Leading Digit Estimate....
    And it has a box saying this:
    Rule for leading digit estimation:
    For each number in the calculations:
    • between 1 and 10, round off to the nearest whole number;
    • between 10 and 100, round off to the nearest multiple of 10; i.e. choose from 10, 20,39..... 100
    • between 100 and 1000, round off to the nearest multiple of 100;
    • i.e. choose from 100, 200, 300,... 1000
    Then for examples they have:
    (a) (= means 'approximately equal to')
    23 x 317 = 20 x 300
    = 6000

    then in small bold writing, it has in brackets (applying the rule 2x3, then add the zeroes)

    then:

    (b) 4250 divided by 23 = 4000 divided by 20
    = 400 divided by 2
    = 200
    then in small bold writing it has in brackets (cancel tens)


    Can you please explain, i haven't got a clue...
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  2. #2
    Senior Member ecMathGeek's Avatar
    Joined
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    Quote Originally Posted by Sazza View Post
    Okay i'm studying about Leading Digit Estimate....
    And it has a box saying this:
    Rule for leading digit estimation:
    For each number in the calculations:
    • between 1 and 10, round off to the nearest whole number;
    • between 10 and 100, round off to the nearest multiple of 10; i.e. choose from 10, 20,39..... 100
    • between 100 and 1000, round off to the nearest multiple of 100;
    • i.e. choose from 100, 200, 300,... 1000
    Then for examples they have:
    (a) (= means 'approximately equal to')
    23 x 317 = 20 x 300
    = 6000

    then in small bold writing, it has in brackets (applying the rule 2x3, then add the zeroes)

    then:

    (b) 4250 divided by 23 = 4000 divided by 20
    = 400 divided by 2
    = 200
    then in small bold writing it has in brackets (cancel tens)


    Can you please explain, i haven't got a clue...
    When you use this method, you are able to "approximate" the answer to these problems. Multiplying 23 x 317 is not very easy (there are a lot of steps), but we can basically figure out how "big" the answer will be if we round 23 to the tens and 317 to the hundreds.

    First, let's round 23 and 317 and multiply these rounded numbers. (~= means "approximately equals"). I assume you know how to round.

    317 ~= 300
    23 ~= 20

    Multiplying these, we get
    . . . . 300
    . . . .x 20
    . . . . 000
    . . . 6000
    . . . 6000

    What this tells me is that 23 x 317 equals something close to 6000. I don't know exactly what it does equal, but the answer should be around 6000. To prove this, let's see what it actually equals:
    . . . . 317
    . . . .x 23
    . . . . 951
    . . . 6340
    . . . 7291

    We approximated the answer was "around" 6000, and the actual answer is 7291. clearly 7291 does not equal 6000, but these answers are reasonably close. That's all we were hoping for. We wanted to figure out approximately what the answer would be to this problem by rounding the numbers.

    We can take a similar approach when dividing and we will get a similarly close approximation for our answer.
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  3. #3
    Super Member

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    Lexington, MA (USA)
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    Hello, Sazza!

    Are you baffled by their hints?


    (a) .23 317 . .20 300 .= .6000

    [apply the rule, 2 3, then add the zeroes]
    When you multiply 20 by 300, do not do it like this:
    Code:
            3 0 0
              2 0
          --------
            0 0 0
          6 0 0
         ---------
          6 0 0 0
    It's correct, but some ten-year-olds will laugh at you.

    The only serious multiplication you did was: .2 3 = 6
    . . All the rest was "zero times something".

    So the answer is: "6 and a bunch of zeros"
    . . Okay, how many zeros?

    Just count them up!
    20 has one zero, 300 has two zeros.
    . . There are three zeros. .(Add them.)

    The answer is: ."6 with three zeros" .= .6000

    See? .You can "eyeball" the problem and get the answer.



    (b) .4250 23 . .4000 20 .= .200

    [cancel tens]

    Once again, you are not expected to divide like this:
    Code:
                  2 0 0
              -----------
          2 0 ) 4 0 0 0
                4 0
                ---
                    0 0
                    0 0
                    ---

    The only real division you did was: .4 2 .= .2
    . . All the rest was "0 divided by 20"

    So the answer is: "2 and a bunch of zeros"
    . . Okay, how many zeros?


    4000 has three zeros. .20 has one zero.
    . . This time we subtract: .two zeros.

    The answer is: ."2 and two zeros" .= .200


    Get the idea?

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  4. #4
    Member Sazza's Avatar
    Joined
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    Australia
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    Arrow

    Yes! But would you have to be able to originally know where the 20 x 300 came from?
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