How can I prove $\frac{m+a}{m+a+l+b}$ is between $\frac{m}{m+l}$ and $\frac{a}{a+b}$?

Question

Let $m,a,l,b \in \mathbb{Z}^{+}$

score 2 · Accepted Answer · answered Jul 27 '16 at 13:15

2

Hint: $$ \frac{a+c}{b+d}-\frac ab=\frac{bc-ad}{b(b+d)} $$ and $$ \frac{a+c}{b+d}-\frac cd=\frac{ad-bc}{d(b+d)} $$

answered Jul 27 '16 at 13:15

robjohn

Hey can you explain more please – Sathasivam K Jul 27 '16 at 14:04
Note that the two differences have opposite signs since $b,d\gt0$. – robjohn Jul 27 '16 at 14:36
@SathasivamK Yours is a special case of a mediant so it has a vivid geometric proof (hint: the slope of the diagonal of a parallelogram lies between the sides). – Bill Dubuque Jul 27 '16 at 16:12

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