Proof to find fraction inbetween to fraction

[Taylor_1989]

I keep getting slightly confused with the algebraic method of finding a fraction between, two other fraction. Here is an example question, I have been doing

Find the fraction between 13/15 and 14/15? I personally convert both to percentages and find the average between the two, the convert back to a fraction. So in this case I did:

13/15 = 0.8666666 = 87%
14/15 = 0.933333 = 93%
87+93 = 180/2= 90
90/100 = 9/10: which I believe is correct

The method I do not understand is the proof behind this method: 13/15+14/15 = 27/30 / 3 top and bottom you get 9/10. But its the proof that confuse me. I will show where I get confused:

I understand this part: a/b < c/d cross multiply ad < bc

This is the part I don't understand: Add ab to both sides ab+ad < ab+bc

why add ab to both sides, where dose this come from?

then factor : a(b+d)<b(a+c)⇒ a/b < a+c/a+b

Then you add cd to both sides, once again why? then you factor out again.

 

[Vorde]

Think of it this way:

You start with 13/15 and 14/15

13/15 is equal to 26/30
and
14/15 is equal to 28/30

This is done by a simple multiplication of two on both the numerator and the denominator.

Looking at your new fractions, it is obvious that 27/30 is in between the two of them, and 27/30 simplifies to 9/10

 

[Mentallic]

This is the part I don't understand: Add ab to both sides ab+ad < ab+bc

why add ab to both sides, where dose this come from?

Because it works. Notice in the next line the factoring worked out such that we can get a/b on its own on the left side.

then factor : a(b+d)<b(a+c)⇒ a/b < a+c/a+b

Just a typo but it should be a/b < (a+c)/(b+d)

Then you add cd to both sides, once again why? then you factor out again.

I guess when you say you add cd to both sides you're talking about

ad&lt;bc

cd+ad&lt;cd+bc

d(a+c)&lt;c(b+d)

\frac{a+c}{b+d}&lt;\frac{c}{d}

Which again work exactly the way we want it to. We've now just shown that \frac{a}{b}&lt;\frac{a+c}{b+d}&lt;\frac{c}{d}

By using algebraic manipulations that were cleverly used to give us the answer we were looking for.
But keep in mind that this value x=\frac{a+c}{b+d} is not always exactly in the middle of a/b and c/d. When b and d are different, it doesn't turn out to be the average of the two fractions.

If you wanted the average of a/b and c/d as your in-between fraction, then you'd have

x=\frac{\frac{a}{b}+\frac{c}{d}}{2}
=\frac{ad+bc}{2bd}

Which is a lot more calculations than the value of x obtained from the proof above.

 

[pwsnafu]

Trivia: the fraction $\frac{a+c}{b+d}$ is called the mediant of $\frac{a}{b}$ and $\frac{c}{d}$.

Further the mediant does not preserve order. Suppose that $\frac{a}{b} < \frac{A}{B}$ and $\frac{c}{d} < \frac{C}{D}$, but it is possible to have $\frac{a+c}{b+d} > \frac{A+C}{B+D}$. This is known as Simpson's paradox.