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If $a, b, c, d, e, f, g, h$ are positive numbers satisfying $\frac{a}{b}<\frac{c}{d}$ and $\frac{e}{f}<\frac{g}{h}$ and $b+f>d+h$, then $\frac{a+e}{b+f} < \frac{c+g}{d+h}$.

I thought it is easy to prove. But I could not. How to prove this? Thank you.

The question is a part of a bigger proof I am working on. There are two strictly concave, positive valued, strictly increasing functions $f_1$ and $f_2$ (See Figure 1). Given 4 points $x_1$, $x_2$, $x_3$ and $x_4$ such that $x_1< x_i$, $i=2, 3,4$ and $x_4> x_i$, $i=1, 2, 3$, let $d=x_2-x_1$, $b=x_4-x_3$ $c=f_1(x_2)-f_1(x_1)$, $a=f_1(x_4)-f_1(x_3)$. And given 4 points $y_1$, $y_2$, $y_3$ and $y_4$ such that $y_1< y_i$, $i=2, 3,4$ and $y_4> y_i$, $i=1, 2, 3$, let $h=y_2-y_1$, $f=y_4-y_3$ $g=f_2(y_2)-f_2(y_1)$, $e=f_2(y_4)-f_2(y_3)$.

Since the functions are concave, we have $\frac{a}{b}<\frac{c}{d}$ and $\frac{e}{f}<\frac{g}{h}$. And I thought in this setting, it is true that $\frac{a+e}{b+f} < \frac{c+g}{d+h}$ even without the restriction $b+f>d+h$.

Figure 1

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  • $\begingroup$ I added a new restriction $b+f>d+h$. Refer to Keith's nice counter example for the original posting. $\endgroup$
    – Sang
    Jul 9, 2015 at 12:24
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    $\begingroup$ Your unlucky numbers are $a,b,c,d,e,f,g,h = 1, 1, 6, 5, 10, 7, 2, 1$. May I ask you what the original purpose of this question is? (I suppose it is something related to Farey series.) $\endgroup$
    – darij grinberg
    Jul 9, 2015 at 12:26
  • $\begingroup$ Hi Darij, it is part of a proof I am struggling with. I tried to describe what I am doing by adding more to the original question. Please refer to the edited question. Thank you. $\endgroup$
    – Sang
    Jul 9, 2015 at 13:32
  • $\begingroup$ This is a great example of something that seems obvious but is actually demonstrably false. In statistics this phenomenon is well-known by the name of Simpson's paradox. $\endgroup$
    – Erick Wong
    Jul 9, 2015 at 14:34

3 Answers 3

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This is false.

For example, $$\frac{1}{3} < \frac{5}{12}, \quad \frac{52}{5} < \frac{11}{1}, \quad \frac{53}{8} > \frac{16}{13}.$$

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  • $\begingroup$ Is there a counterexample that uses fractions in lowest terms? $\endgroup$
    – G Tony Jacobs
    Jul 8, 2015 at 16:06
  • $\begingroup$ @GTonyJacobs I've edited my answer. $\endgroup$
    – Keith
    Jul 8, 2015 at 16:09
  • $\begingroup$ Nice counterexample. $\endgroup$
    – vadim123
    Jul 8, 2015 at 16:15
  • $\begingroup$ It is a very nice counterexample. Is there a counter example that satisfies an added restriction, $b+f>d+h$? Or the statement is true with the added restriction? $\endgroup$
    – Sang
    Jul 9, 2015 at 12:20
  • $\begingroup$ @Sang The extra restriction is irrelevant: just rescale $a,b,e,f$ by the same large integer and you'll get $b+f > d+h$ without changing any of the values being compared. You could then add $1$ to $b$ and $f$ and have a decent chance of the resulting fractions being in lowest-terms (certainly high enough that you could just pick a larger integer that works): these are strict inequalities so they are tolerant of minute adjustments. $\endgroup$
    – Erick Wong
    Jul 9, 2015 at 14:31
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The updated question (with the additional constraint $b+f>d+h$) is also false. For example,

$\frac{1}{1}<\frac{3}{2}$ and $\frac{9}{4}<\frac{5}{2}$, but $\frac{1+9}{1+4} = \frac{10}{5} = \frac{8}{4} = \frac{3+5}{2+2}$.

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As

$$\frac{a+e}{b+f} < \frac{c+g}{d+h}$$ $$(a+e)(d+h) < (c+g)(b+f)$$ $$ad+eh+ah+ed < cb+fg+cf+gb$$ here put $ah+ed = x$ and $cf+gb = y$. Now we have $$ad+eh+x < bc+fg+y$$ Now as the given $$\frac{a}{b}<\frac{c}{d} \implies { ad<bc}$$ and $$\frac{e}{f}< \frac{g}{h} \implies{eh<fg}$$ so by adding both the above eqs $$ad+eh<bc+fg$$ and by this condition we have $$ad+eh+x<bc+fg+y$$

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    $\begingroup$ How do you justify the last step? It is certainly true if $x<y$ (in particular for $\frac{a}{b}<\frac{g}{h}$ and $\frac{c}{d}<\frac{e}{f}$), but we cannot assume that. $\endgroup$
    – Klaus Draeger
    Jul 9, 2015 at 12:50

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