Proof regarding fractions

[captainquarks]

Not sure if this is the correct place to put this question, but here it goes (sorry about the vague title, but not sure how to describe it):

We are asked to consider two rational fractoins, for example:

a/b & x/y

We are now asked to do the following:

(a + x)/(b + y)

Now, we are told that this new, rational expression lies in the range between our initial rational fractionss..

We are asked to prove that this is generally true, with our constants as integers of course:

e.g. 7/3, 9/2 => (7+9)/(3+2) = 16/5, which is between the originals.

I started by doing basic inequalities

x/y < (a + x)/(b + y) < a/b

I split these up:

x/y < (a + x)/(b + y)
x(b + y) < y(a + x)
bx + xy < ay + yx
bx < ay

#

(a + x)/(b + y) < a/b
b(a + x) < a(b + y)
ba + bx < ab + ay
bx < ay

I have no idea (or havn't found that spark yet) as to how, or where to solve this problem, which is very frustrating indeed. Does anyone have any insight into what I'm doing? Input would be greatly appreciated.

 

[DonAntonio]

Not sure if this is the correct place to put this question, but here it goes (sorry about the vague title, but not sure how to describe it):

We are asked to consider two rational fractoins, for example:

a/b & x/y

We are now asked to do the following:

(a + x)/(b + y)

Now, we are told that this new, rational expression lies in the range between our initial rational fractionss..

We are asked to prove that this is generally true, with our constants as integers of course:

e.g. 7/3, 9/2 => (7+9)/(3+2) = 16/5, which is between the originals.

I started by doing basic inequalities

x/y < (a + x)/(b + y) < a/b

I split these up:

x/y < (a + x)/(b + y)
x(b + y) < y(a + x)
bx + xy < ay + yx
bx < ay

#

(a + x)/(b + y) < a/b
b(a + x) < a(b + y)
ba + bx < ab + ay
bx < ay

I have no idea (or havn't found that spark yet) as to how, or where to solve this problem, which is very frustrating indeed. Does anyone have any insight into what I'm doing? Input would be greatly appreciated.

Shouldn't the condition "positive integers" be added? For example, we have $$\frac{1}{-4}<\frac{3}{2}\,\,\,but\,\,\, \frac{3+1}{-4+2}=-2<\frac{1}{-4}...$$.
Now, if all the numbers are natural ones (no zero included), then we have $$\frac{x}{y}<\frac{a}{b}\Longrightarrow \frac{x+a}{y+b}<\frac{a}{b}\Longleftrightarrow bx+ab<ay+ab\Longleftrightarrow bx<ay$$ and this last inequality is the very same we started with.

Something similar can be done to show the other side's inequality

DonAntonio

 

[mathman]

bx < ay implies x/y < a/b (assuming all integers involved are positive).

So start with assuming x/y < a/b and work your derivations in reverse order.

 

[captainquarks]

Thanks DonAntonio and mathman for your replies and help. I think I get where you're coming from, but I'm not the best Mathematician in the world (Below 1st year university level Id say)...

A little more explanation/algebra would be great...

I was thinking, that, if I got my inequalities as i did, then if i work back, i obviously get the same inequality, does the answer lie in the crux on this? I just can't seem to express myself mathematically as well as others

 

[captainquarks]

Does anyone else have any ideas? I just can't seem to get my head around it =(

 

[Office_Shredder]

You wrote

x/y < (a + x)/(b + y)
x(b + y) < y(a + x)
bx + xy < ay + yx
bx < ay

Now the key to these calculations is that they are all reversible - i.e. not only can you go from bx + xy < ay + yx to bx < ay, but from bx < ay you can get bx + xy < ay + yx

Now we have a,b,x,y with x/y < a/b. So from here we can derive that bx<ay. Now we take all the calculations you do and reverse them
bx<ay
bx + xy < ay + yx
x(b + y) < y(a + x)
x/y < (a + x)/(b + y)

And this is a proof that if x/y < a/b, then x/y<(a+x)/(b+y)

 

[DonAntonio]

Does anyone else have any ideas? I just can't seem to get my head around it =(

More ideas?! You've already been shown by at least two different people the very solution to your problem. If you still

don't get it then either you don't know the basic properties and operations with fractions yet, and then you need to

read about this in some basci H.S. algebra book, or else something's definitely beyond your understanding capabilities right now (and

this happens to us all at some point, don't overworry) and thus you need to approach somebody (preferable mathematician) who

explain you this stuff personally. After you get it you'll laugh at how easy it is...just as most of us had at some point.

DonAntonio

 

[captainquarks]

I do get it now, i was just expecting it to be more rigorous that's all, as the other stuff I've been looking at is induction etc, i was expecting something along the same lines, that's all. I can do induction reasonably well etc. however, thanks for your time =)