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I am new to mathematics. When I was doing course work on fractions. I learned something like follows

$(2 + 5) / (4 + 10)$

learned a common pattern

$(X/2 + Y/2)/ (X + Y)$ or $(X + Y)/ (2(X + Y))$

I plugged in some number and tested, it seems to be right.

  • To which field(number theory, arithmetic) this kind of question belongs to?
  • Is this a conjecture? theorem?
  • How could I proof such a simple equation hold util infinity?
user158
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    Do you mean $(x/2 + y/2)/(x+y)$ (and $(2+5)/(4+10)$)? The parenthesis are important: they determine the order of operations! – Magdiragdag Oct 16 '17 at 06:33
  • Your parenthesing looks incorrect. Your question belongs to elementary arithmetic and is a simple consequence of associativity and distributivity of multiplication and division. Technically speaking it's a theorem, but a very "immediate" one. It holds for any numbers such that $X+Y\ne0$, and the expression simplifies to just $1/2$. –  Oct 16 '17 at 06:34
  • @Magdiragdag corrected – user158 Oct 16 '17 at 06:51
  • @YvesDaoust Can you elaborate "a simple consequence of associativity and distributivity of multiplication and division." – user158 Oct 16 '17 at 06:53
  • @user158: I did ! –  Oct 16 '17 at 06:55
  • @YvesDaoust thanks – user158 Oct 16 '17 at 06:57
  • @user158 Your parenthesis are still incorrect in $2 + 5/4 + 10$ and in $(X+Y)/2(X+Y)$. The last one is evaluated left to right, but you mean $(X+Y)/(2(X+Y))$. – Magdiragdag Oct 16 '17 at 06:59
  • @Magdiragdag thanks again – user158 Oct 16 '17 at 07:07

1 Answers1

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$$\frac{\dfrac X2+\dfrac Y2}{X+Y}=\frac{\dfrac12(X+Y)}{X+Y}=\dfrac12\frac{X+Y}{X+Y}=\frac{X+Y}{2(X+Y)}=\frac12.$$

Justifications:

$$ab+ac=a(b+c),\\\frac ab\frac cd=\frac{ac}{bd}.$$