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The question is mostly in the title. If I have a numerical value for a ratio $q_1 = x_1/x_2$, how do I compute $q_2 = x_2 / (x_1 + x_2)$?

For example, if $x_1 = 1$ and $x_2 = 8$, we would be given $q_1 = 0.125$ (without explicitly being given $x_1$ and $x_2$), and we would want to find $q_2 =8 / (8+1) = 0.88888...$

How does one generate $q_2$ from $q_1$ without access to $x_1$ and $x_2$?

Asimov
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Jeff
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    $$\frac{1}{q_2} = \frac{x_1}{x_2} + 1 = q_1 + 1$$

    Inverting both sides gives that $q_2 = 1 / (q_1 + 1)$.

     –  Jun 04 '14 at 22:02
  • @user61527: that should be an answer. Please post it as such so we have one. Thanks – Ross Millikan Jun 04 '14 at 22:04
  • In your line For example, it should be q1=0.125 – Ross Millikan Jun 04 '14 at 22:04
  • @RossMillikan Fair enough - I tend to dislike answering if it's a one or two line answer (and I see that Jika has already left a longer answer). –  Jun 04 '14 at 22:18
  • @user61527: I think one or two lines is fine if it answers the question, which yours does. But we do have an answer with a few more steps, so no problem. – Ross Millikan Jun 04 '14 at 22:19

1 Answers1

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You have $$q_1=\dfrac{x_1}{x_2},$$

and $$q_2=\dfrac{x_2}{x_1+x_2},$$

Divide the numerator of $q_2$ and the denominator of $q_2$ by $x_2$. You will get:

$$q_2=\dfrac{\dfrac{x_2}{x_2}}{\dfrac{x_2+x_1}{x_2}},$$

which, as you can see equals:

$$q_2=\dfrac{1}{1+\dfrac{x_1}{x_2}}=\dfrac{1}{1+q_1}.$$

Jika
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