Math Fraction Question: Has anyone seen this not work?

Question

If you are looking for a fraction to fall between two other fractions, you can add the denominators and numerators, and the new fraction consistently falls between those. For example, 8/15 and 2/3, then 8+2=10, and 15+3=18, so you get 10/18, or 5/9 which is between those. How does it work? We need a counter-example, if there is one.

Answer 1

we need to prove
if a/b < c/d ---1
(1) => ad < bc
then
a/b < (a+c)/(b+d) < c/d
first i will prove a/b < (a+c)/(b+d)
ab = ba
ad < bc
ab + ad < ba + bc
a(b+d) < b(a+c)
a/b < (a+c)/(b+d)
Now (a+c)/(b+d) < c/d
cd = dc
ad < bc
cd + ad < dc + bc
(a+c) d < c(b+d)
=> (a+c)/(b+d) < c/d
Thus
a/b < (a+c)/(b+d) < c/d

Answer 2

Suppose your original fractions are a/b and c/d with a/b > c/d. That inequality is equivalent to the statement ad > bc (if you cross multiply).

So how does the fraction (a + c)/(b + d) compare?

Compare to a/b:
a/b ? (a+c)/(b+d)
a(b+d) ? b(a + c)
ab + ad ? ab + bc
Since we know that ad > bc, then working backwards we know that a/b > (a+c)/(b+d). Similar logic will show that (a+c)/(b+d) > c/d.

This should work for any positive fraction. Not sure about negatives. Good luck!

Answer 3

1/10 + 1/5 = 1/15 which is <1/10

Answer 4

that doesnt work olga

1/1 + 1/1000