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Math Help - [SOLVED] two digit number

  1. #1
    troubled kid
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    Unhappy [SOLVED] two digit number

    plzzz help me! im so dumb i hate maths wit passion!!!
    1. dan the detective looked for a number. he found a teo-digit number less than 50. the sum of its digits was 12. their diffrence was 4. what number did Dan find?

    some 1 can u tel me how im meant to do this! plzzzzzzzz
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  2. #2
    MHF Contributor
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    Dan must be from outer space. They have teo-digit numbers over there.
    If Dan found only a two-digit number, then he may be from our earth.

    Two-digit number: xy ---not x*y, but (10x +y).
    xy < 50 ----(i)
    x+y = 12 ---(ii)
    x -y = 4 -----(iii), if x is greater than y.

    Add (ii) and (iii),
    2x = 16
    x = 16/2 = 8
    Substitute that into (ii),
    x +y = 12
    y = 12 -8 = 4
    So xy = 84, which is not less than 50, hence, wrong.

    So we try the next option for (iii).
    y -x = 4 ----(iiia), y is greater than x.
    y = 4 +x
    Substitute that into (ii),
    x+y = 12 ----(ii)
    x +(4+x) = 12
    x +4 +x = 12
    x +x = 12 -4
    2x = 8
    x = 8/2 = 4
    Substitute that into (ii),
    4 +y = 12
    y = 12 -4 = 8
    Then, xy = 48, which is less than 50.

    Therefore, Dan found the number 48. -----answer.
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  3. #3
    Junior Member Euclid Alexandria's Avatar
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    Lightbulb

    It is also possible to do this in your head, but the steps should be more limited. We can look at this like a puzzle instead of just a math problem, just like a detective would. So we will try a natural, trial-and-error approach, just like you would by trying different pieces of a jigsaw puzzle.

    First, we want a two-digit number, and we want the sum of its digits to be 12. The easiest number to think of is 66.

    6 + 6 = 12
    6 - 6 = 0

    But the problem also demands that their difference must be 4. Oops!

    To keep the sum of the digits at 12, let's try something new. We will make one digit larger at the same time that we are making the other digit smaller.

    Since we are doing this in our head, we will make it easier by doing it one digit at a time. So if we make the first 6 one digit larger, we will make the other 6 one digit smaller.

    7 + 5 = 12
    7 - 5 = 2


    This still equals 12, but we only have a difference of 2! Let's try it again. One digit larger, one digit smaller.

    8 + 4 = 12
    8 - 4 = 4


    We have finally found a two digit number, 84, that has a sum of 12 and a difference of 4. But the sum of these two numbers needs to be below fifty! Oops!

    But I think we're on to something. Taking a look at the number 84, we see that if we just switch the numbers around, the sum will stay the same, and the difference will still be 4, even though it will become negative. The two digit number then becomes 48, which is less than 50.

    4 + 8 = 12
    4 - 8 = -4


    So, Dan the detective must have found the number 48.
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  4. #4
    Newbie krassi_holmz's Avatar
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    Smile The easiest way

    Let the two digits are x and y and x>y. Then
    x+y=12 and x-y=4 =>
    4+2y=12
    2y=8
    y=4
    x=8
    We have 2 opportunities: 48 and 84, but the number is less than 50 so it's 48.
    Cool
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  5. #5
    Junior Member Euclid Alexandria's Avatar
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    Exclamation

    A couple of things about that approach don't seem right to me (I admit that I'm not one of the math geniuses on the forum). First, wouldn't the 4 have to be negative? Since the only option we have under 50 is 48, that means we're subtracting 8 from 4 as well as adding it. Right?

    4 + 8
    4 - 8


    Secondly, the reason I approached the problem intuitively rather than systematically (if that's the right way to put it), is because the problem was posed by a nine year-old. A nine year-old who hates "maths with passion" to boot.

    I responded with that sort of audience in mind. Would a nine year-old in that position understand all the variables being thrown around here?
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  6. #6
    Newbie krassi_holmz's Avatar
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    Agreement

    First, I don't agree with you. The original question means
    x+y=12 and |x-y|=4. But in condition x>y => x-y>0, so x-y=4. We may take this condition because we don't know which digit comes first- x or y, so the condition means "first digit is greater than second or second is greater than first", because x may be and first and second digit.

    Secondly, I fully agree with you.
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  7. #7
    Junior Member Euclid Alexandria's Avatar
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    So, what you're saying is that since no condition is explicitly stated, we can assume 84 (8 + 4 and 8 - 4) for both the sum and the difference, and then switch the numbers around so that we have 48? Right?

    In a way, that makes more sense to me, because the question never said anything about a negative number. But in that case, I think the question should have stated clearly that the digits of the answer are interchangeable just like the digits in the problem. After all, as you can see from my solution, someone solving it intuitively (such as a nine year-old, perhaps) may be likely to approach it with the same logic I approached it with.
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  8. #8
    Newbie krassi_holmz's Avatar
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    Yes, I wanted to say this. You've understood.
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  9. #9
    Junior Member Euclid Alexandria's Avatar
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    Lightbulb

    I now see that I made an incorrect assumption about the problem.

    "Dan the detective looked for a number. He found a two-digit number less than 50. The sum of its digits was 12. Their diffrence was 4. What number did Dan find?"

    For my intuitive approach to be correct, the problem would need to request something convoluted, like "The sum of its digits, in the same order as the digits in the number that Dan found, was 12, and their difference was -4." Now I see that this part of the problem was only talking about the two separate digits involved, 8 and 4, and not specifically the number that Dan found.

    Sometimes it's good to walk away from a problem for a while, then look at it again with fresh perspective and fewer assumptions. The problem was stated clearly, but I didn't understand it clearly on first (or second, or third) glance.
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  10. #10
    Senior Member TriKri's Avatar
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    Your'e class are solving equation systems? If your'e only nine you must be in a school for extra talented kids. Sweden wish you good luck with the mathematics in the future!
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  11. #11
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    Step by step analysis of the problem

    1. If the two digit number is less than 50. Means that the first digit is either 1, 2, 3 or 4.

    2. The sum of the two digit must be 12
    So the first digit cannot be 1 and 2
    Because 1 + 11 give you 12 and 2 + 10 give you 12. 11 and 10 are double digit.

    We are left with 3 and 4 for the first digit.

    Option 1->
    If first digit is 3:
    3 + 9 = 12

    Option 2->
    If first digit is 4:
    4 + 8 = 12

    3. Difference of the two digit is 4
    For option 1.
    Difference of 3 and 9 is 6. So this is wrong.

    For option 2.
    Difference of 4 and 8 is 4. So the correct answer is 48


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