Math Proportion problem..a/b=c/d?

  1. If a, b, and c are numbers such that a/b=3, b/c=7, then a+b/b+c is equal to what?
    Please answer this question in a detailed way if you can, and also please provide me with the basic concept this problem originated from.
    Thank you

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    Answer by John

    a/b=3 b/c=7
    a/b=3 49/7=7 for the reason of 7*7
    147/49=3 for the reason of 49*3

  2. Show that (a, b – ta) = (a, b).
    where (a,b) is the greatest common divisor of a and b and a and b do not equal 0

    Answer by kb
    Let d = (a, b) and e = (a, b – ta).

    (i) Since d = (a, b), we have d | a and d | b.
    Thus, d | (-ta) and so d | (b – ta).
    Since d | a and d|(b – ta), we have that e | d.

    (ii) Since e = (a, b – ta), we have e | a and e | (b – ta).
    So, e | ta ==> e | (ta + (b – ta)) = b.
    Since e | a and e | b, we have that d | e.

    By (i) and (ii), we conclude that d = e (since the gcd is positive).

    I hope this helps!

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