Why a(pt=proportional to) pt b and a pt c implies a pt bc?

[Rishabh Narula]

hmm.very thankful to everyone who answered but i think where I am actually getting stuck is nothing high level.im just not getting why a proportional to b and a proportional to c implies a proportional to bc .i get it that if b changes by factor x then a changes by factor x and at the same time if c changes by y then that a.x would change by y resulting a net change a into a.x.y and also i mean yeah when you see a proprtional to bc it is evident that b,c getting changed by x,y makes a change by xy.but i want some more derived proof or something i guess.

 

[blue_leaf77]

##a## proportional to ##b## means ##a=K_1b## and ##a## proportional to ##c## means ##a=K_2c##. Combining the last two equations, you get ##K_1b=K_2c## or ##\frac{K_1}{K_2}=\frac{c}{b}## valid for all ##b## and ##c##. This equation is satisfied by ##K_1=K_3c## and ##K_2=K_3b## with ##K_3## being non-zero. Therefore ##a=K_3bc##.

 

[Rishabh Narula]

##a## proportional to ##b## means ##a=K_1b## and ##a## proportional to ##c## means ##a=K_2c##. Combining the last two equations, you get ##K_1b=K_2c## or ##\frac{K_1}{K_2}=\frac{c}{b}## valid for all ##b## and ##c##. This equation is satisfied by ##K_1=K_3c## and ##K_2=K_3b## with ##K_3## being non-zero. Therefore ##a=K_3bc##.

##a## proportional to ##b## means ##a=K_1b## and ##a## proportional to ##c## means ##a=K_2c##. Combining the last two equations, you get ##K_1b=K_2c## or ##\frac{K_1}{K_2}=\frac{c}{b}## valid for all ##b## and ##c##. This equation is satisfied by ##K_1=K_3c## and ##K_2=K_3b## with ##K_3## being non-zero. Therefore ##a=K_3bc##.

this helped a lot.thank you for your time.