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Math Help - Find the Value

  1. #1
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    Find the Value

    The following 7 numbers have the same mean, median, and m ode. If 23 is not one of the numbers, find the value of x + y.

    39, 25, 19, 19, 25, x, y
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  2. #2
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    Quote Originally Posted by MathMage89 View Post
    The following 7 numbers have the same mean, median, and m ode. If 23 is not one of the numbers, find the value of x + y.

    39, 25, 19, 19, 25, x, y
    The "mode" exists.
    Meaning there is a number which appears most often.
    Thus, there are three possibilities:
    1)Mode is 25 ( x=25).
    2)Mode is 19 ( x=19).
    3)Mode is 39 ( x=y=39).

    It cannot be case #3 because then the mean is 29.2 a violation of initial conditions of problem.

    If #2 then the mean must be 19.
    Meaning,
    \frac{39+3\cdot 19+2\cdot 25+y}{7}=19
    \frac{146+y}{7}=19
    y=-13
    Checking the mode we find that this works.

    If #1 then the mean must be 25.
    Meaning,
    \frac{39+3\cdot 25+2\cdot 19+y}{7}=25
    \frac{152+y}{7}=25
    y=23
    But the problem says it is not amongs the numbers.

    Thus, we have determined the unique numbers the work.
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  3. #3
    Junior Member AlvinCY's Avatar
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    The following 7 numbers have the same mean, median, and mode. If 23 is not one of the numbers, find the value of x + y.

    39, 25, 19, 19, 25, x, y

    For this question, I'll assume that the mode is infact unique, that is, you can't have more than one more, otherwise, you'll need 2 means, and that's impossible with a set of data

    Rearrange to give 19, 19, 25, 25, 39... and x and y we don't know where to place yet. But no matter where we put it... the median has to br 19 or 25 (just move x and y around and use some logic, you'll see that this is true)

    Mean = \frac{39 + 25 + 19 + 19 + 25 + x + y}{7}

    We want this to be a nice whole number, so 39 + 25 + 19 + 19 + 25 + x + y must be a multiple of 7, 39 + 25 + 19 + 19 + 25 = 127, trial and error the next few mulitples of 7, i.e. 133, 140, 147, 154, 161, 168, 175, 182, 189, 196... we know x + y = 6, 13, 20, 27, 34, 41, 48, 55, 62, 69...

    At this stage we still don't know what to expect x and y to be, but we're certainly getting closer... we want to achieve a unique mode, seeing we already have two of 19 and two or 25, we can safely say that one of x or y = 19 or 25. We've already narrowed it down to 2 cases, mode = 19 and mode = 25 (seeing the median can only be 19 or 25)

    Test 1: If mode = 19, then mean = 19 (as the question suggests), so the sum of the scores must be 19 \times 7 = 133 suggesting that x + y = 6, and we want either x or y to be 19, so x = 19, y = -13, vice versa... I know this will raise a few questions in that y is negative, but it never said in the question that these numbers can't be negative, all they ask for is the sum. And I'll show you why in this following case ~

    Test 2: If mode = 25, the mean = 25 (as the question suggests), so the sum of the scores must be 25 \times 7 = 175 suggesting that x + y = 48, to achieve a unique mode we need either x or y to be 25:

    When x = 25, y = 23, and vice versa, this violates the rule that one of the numbers aren't 23.

    So we know the mean = median = mode = 19 when the scores are:

    -13, 19, 19, 19, 25, 25, 39

    Check:

    Mean = \frac{-13 + 19 + 19 + 19 + 25 + 25 + 39}{7} = 19
    Mode = 19 (19 being the most frequent) and;
    Median = 19, as 19 is the middle score.

    So we can conclude that x + y = 6.
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