Gradients between two gradients?

[apple2357]

Is there a simple explanation of why this is the case?

Suppose i have two fractions e.g 5/12 and 3/5

If i wanted to create a fraction between the two i can simply add the numerators and denominators to get 8/17.

This appears to work all the time. I haven't proved it using algebra but i feel there should be a simple explanation, maybe visually but i am stuck. Can anyone help? I am happy to accept there isn't an intuitive explanation but would like to rule it out!

 

[pka]

Is there a simple explanation of why this is the case?
Suppose i have two fractions e.g 5/12 and 3/5

\(\displaystyle \text{LCM}(5,12)=60\) so \(\displaystyle \frac{5}{12}=\frac{25}{60}~\&~\frac{3}{5}=\frac{36}{60}\).
What is between?

 

[Dr.Peterson]

Your calculation is something called the mediant of the fractions. Wikipedia states the inequality you are asking about, with a proof.

If you look further down that page, you'll see a graphical version that makes it almost immediately obvious, in terms of vectors <a,b> and <c,d>, whose slopes are your two fractions. The sum of the vectors corresponds to the mediant, and its slope is clearly between the two fractions you start with.

 

[apple2357]

How does that help explain why adding numerators and denominators gives a fraction between the two?

 

[apple2357]

Your calculation is something called the mediant of the fractions. Wikipedia states the inequality you are asking about, with a proof.

If you look further down that page, you'll see a graphical version that makes it almost immediately obvious, in terms of vectors <a,b> and <c,d>, whose slopes are your two fractions. The sum of the vectors corresponds to the mediant, and its slope is clearly between the two fractions you start with.

Thanks. Thats exactly what i was looking for!

 

[Deleted member 4993]

Is there a simple explanation of why this is the case?

Suppose i have two fractions e.g 5/12 and 3/5

If i wanted to create a fraction between the two i can simply add the numerators and denominators to get 8/17.

This appears to work all the time. I haven't proved it using algebra but i feel there should be a simple explanation, maybe visually but i am stuck. Can anyone help? I am happy to accept there isn't an intuitive explanation but would like to rule it out!

Yes ... that will be true for every pair of fraction:

\(\displaystyle If \ \ \ \ \frac{A}{B} \gt \frac{C}{D} \ \ \ \ then \ \ \ \ \frac{A}{B} \gt \frac{A+C}{B+D} \gt \frac{C}{D}\)

It can be proven relatively easily. You should try to prove it - I did.

Nice observation though - I did not see this before.

("You can observe a lot, if you just look" - paraphrasing Yogi Berra)