- #1

RChristenk

- 49

- 4

- Homework Statement
- Show that the ratio ##x+y:x-y## is increased by subtracting ##y## from each term.

- Relevant Equations
- ##x=ky##

##x+y:x-y=\dfrac{x+y}{x-y} \tag1##

Subtract ##y## from each term:

##x:x-2y=\dfrac{x}{x-2y} \tag2##

Assume ##k=\dfrac{x}{y} \Rightarrow x=ky##

##(1)= \dfrac{ky+y}{ky-y}, (2)= \dfrac{ky}{ky-2y}##

Subtract ##(1)## from ##(2)## since we are told by the problem statement ##(2)## is bigger:

##\dfrac{(ky)(ky-y)-(ky+y)(ky-2y)}{(ky-y)(ky-2y)} \Rightarrow \dfrac{k^2y^2-ky^2-(k^2y^2-2ky^2+ky^2-2y^2)}{k^2y^2-2ky^2-ky^2+2y^2} \Rightarrow \dfrac{2y^2}{k^2y^2-3ky^2+2y^2}##

##\Rightarrow \dfrac{2}{k^2-3k+2} \Rightarrow \dfrac{2}{(k-2)(k-1)}##

For ##1<k<2; \dfrac{2}{(k-2)(k-1)}<0## and ##\dfrac{x+y}{x-y}>\dfrac{x}{x-2y}##

For ##k<1## and ##k>2##; ##\dfrac{x+y}{x-y}<\dfrac{x}{x-2y}##

Question: The key to solving this problem was assuming ##k=\dfrac{x}{y} \Rightarrow x=ky##. I know how to plug and chug (obviously), but my question is why is this valid? How does one know ##x## varies proportionally with ##y##? Because ##x## and ##y## could be anything, there's no guarantee they vary proportionally. What are the mathematical rules and assumptions that make this work? Thanks.

Subtract ##y## from each term:

##x:x-2y=\dfrac{x}{x-2y} \tag2##

Assume ##k=\dfrac{x}{y} \Rightarrow x=ky##

##(1)= \dfrac{ky+y}{ky-y}, (2)= \dfrac{ky}{ky-2y}##

Subtract ##(1)## from ##(2)## since we are told by the problem statement ##(2)## is bigger:

##\dfrac{(ky)(ky-y)-(ky+y)(ky-2y)}{(ky-y)(ky-2y)} \Rightarrow \dfrac{k^2y^2-ky^2-(k^2y^2-2ky^2+ky^2-2y^2)}{k^2y^2-2ky^2-ky^2+2y^2} \Rightarrow \dfrac{2y^2}{k^2y^2-3ky^2+2y^2}##

##\Rightarrow \dfrac{2}{k^2-3k+2} \Rightarrow \dfrac{2}{(k-2)(k-1)}##

For ##1<k<2; \dfrac{2}{(k-2)(k-1)}<0## and ##\dfrac{x+y}{x-y}>\dfrac{x}{x-2y}##

For ##k<1## and ##k>2##; ##\dfrac{x+y}{x-y}<\dfrac{x}{x-2y}##

Question: The key to solving this problem was assuming ##k=\dfrac{x}{y} \Rightarrow x=ky##. I know how to plug and chug (obviously), but my question is why is this valid? How does one know ##x## varies proportionally with ##y##? Because ##x## and ##y## could be anything, there's no guarantee they vary proportionally. What are the mathematical rules and assumptions that make this work? Thanks.