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This is the problem that I am having trouble with for my test review. I am completely blank and don't know what it is asking for. Can you please guide me step by step. For example: Why did constant $k$ appear all of a sudden?

$a$ varies directly with $b$

Which of these equations could represent the relationship between $a$ and $b$?

$a$ varies directly with $b$ if $a=k\cdot b$ for some constant $k$

If you divide each side of this expression by $b$, you get $\displaystyle\frac ab=k$ for some constant $k$.

$\displaystyle\frac ab=\frac12$ fits this pattern, with $k=\displaystyle\frac12$


  • $a=\frac12-b$
  • $\frac12\cdot\frac1a=b$
  • $2\cdot\frac1a=b$
  • $\frac12\cdot a=\frac1b$
  • $\frac ab=\frac12$
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    Well, I'm not sure why I should take time to answer your question if you think it doesn't really matter whether you put in the minimal effort of making your question correspond to what you're asking about; but I did.2012-05-03

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"$a$ varies directly with $b$" basically says "if you double $b$, then $a$ is doubled; if you multiply $b$ by $10$, then $a$ gets multiplied by $10$", etc.; the two are in direct proportion to each other. The constant $k$ is the proportionality factor. For instance, if the values $1,2,3,4$ for $b$ correspond to values $2,4,6,8$ for $a$, then the proportionality constant is $k=2$, whereas if the values $1,2,3,4$ for $b$ correspond to values $5,10,15,20$ for $a$, then the proportionality constant is $k=5$. I suggest that you play around with the equations offered as choices on the right and convince yourself that the last one is the only one that has the above property. The explanation on the lower left explains that in terms of formulas.

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    @Backtrack: No worries; you're welcome.2012-05-03