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Math Help

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Yes it was answered previously but I am sorry that I did not follow the pattern. I was told that the answer that I submitted was incorrect. Please help if you can. Thank you so very much.

A three-digit number increases by 9 if we exchange the second and third digits. The same three-digit number increases by 90 if we exchange the first and second digits. By how much will the value increase if we exchange the first and third digits?

Thank you

  • Math Help -

    in the original number,
    let the unit digit be a,
    the tens digit be b
    and the hundred digit be c
    the appearance of our number is cba
    then the value of our number is 100c + 10b + a

    case1: interchange 2nd and 3rd digit
    number looks like cab
    value of our new number is 100c + 10a + b
    so 100c + 10a + b - (100c + 10b + a) = 9
    9a -9b = 9
    a-b = 1 , (#1)

    case2: interchange the 1st and 2nd digit
    appearance of new number is bca
    value of new number is 100b + 10c + a
    so 100b+ 10c + a - (100c + 10b + a) = 90
    90b -90c = 90
    b - c = 1 , (#2)

    case3: interchange 1st and 3rd
    appearance of new number is abc
    value of new number is 100a + 10b + c

    change in value = 100a + 10b + c - (100c + 10b + a)
    = 99a - 99c
    = 99(a-c)

    but if we add #1 and #2
    we get
    a - c = 2

    so 99(a-c) = 99(2) = 198

    A little know feature of the above is the following "math trick"
    1. Pick any 3 digit number, all different and the hundreds digit greater than the unit digit
    2. reverse the digits and subtract the numbers. If you get a 2 digit result, insert a 0 in the hundred place
    3. reverse your subtraction answer and add the last two results,
    4. You will always get 1089



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