Fraction rules A/B/C vs B/C/A

Question

I have all my fraction rules written down, but these two confuse me...

$$ \frac{\;\;A\;\;}{\frac{B}{C}} \equiv{\frac{\;AC\;}{B}} $$

and

$$ \frac{\frac{B}{C}}{\;\;A\;\;} \equiv{\frac{\;B\;}{CA}} $$

These do not give the same answers, and I am wondering what the rules are for using them? Are $$A>B>C$$ and $$B>C>A$$ the two conditions?

It's just what the convention for the meaning of the notation is. — Michael McGovern, Jan 29 '18 at 02:02
ok so i just get to pick whichever one helps me solve my algebra, even though they do not give the same answer? — user2355058, Jan 29 '18 at 02:04
@MichaelMcGovern Unfortunately, your comment is mistaken - one of the two rules is wrong as written. — , Jan 29 '18 at 02:06
@user2355058 why did you choose \frac{}{\cfrac{}{}} instead of \frac{}{\frac{}{}}? — Michael McGovern, Jan 29 '18 at 02:11
@user2355058: you absolutely do not want them the same size. You want one fraction bar to be larger to show which division is last. That is the source of your problem. — Ross Millikan, Jan 29 '18 at 02:16
And that part of the problem: the variation in size is a way of designating the grouping. — Michael McGovern, Jan 29 '18 at 02:16
@user2355058 if you want to be understood, you need to write clearly. — Michael McGovern, Jan 29 '18 at 02:25
Thanks -- I guess I just need to go retake algebra 1 and pre algebra. wish me luck — user2355058, Jan 29 '18 at 02:36

score 6 · Accepted Answer · answered Jan 29 '18 at 02:15

6

You may have been confused by the implied parentheses in the size of the fraction bars. Your first rule is $$\frac A{\left(\frac BC\right)}=\frac {AC}B$$ which you can establish by multiplying the fraction by $\frac CC$. You have written the second the same way on the left, $$\frac B{\left(\frac CA\right)}=\frac {B}{CA}$$ which is incorrect. I suspect you meant to write $$\frac {\left(\frac BC\right)}A=\frac {B}{CA}$$ which is correct and can be established by multiplying by $\frac CC$. The only clue in what you wrote is the size of the fraction bars and your top one is the largest in both cases. It is hard to see that in my browser, but the strucure of your MathJax makes it clear.

answered Jan 29 '18 at 02:15

Ross Millikan

374,822

His MathJax is not clear anyway. – Weijun Zhou Jan 29 '18 at 02:17
@WeijunZhou: I looked at the source and saw how the fractions were specified – Ross Millikan Jan 29 '18 at 02:18
The bigger division is simplified last, am I correct? – Yash Jain Jan 29 '18 at 02:19
@RossMillikan This is what I see: $$ \frac{A}{\cfrac{B}{C}} \equiv{\frac{AC}{B}} $$
and

$$ \frac{B}{\cfrac{C}{A}} \equiv{\frac{B}{CA}} $$. I don't see any difference.
– Weijun Zhou Jan 29 '18 at 02:19
1

This is just another variation on the "Does a/bc mean (a/b)c or a/(bc)?" question. – Michael McGovern Jan 29 '18 at 02:22
@YashJain Yes, you are correct. – 4yl1n Apr 12 '20 at 21:09

score 2 · Answer 2 · answered Jan 29 '18 at 02:07

2

No, there are no such "rules" for using them, $A>B>C$ or $B>A>C$ don't have to be conditions. To make fractions like these seem simpler, think of division as the "opposite" of multiplication. I'll show you what I mean.

$\cfrac{A}{\frac{B}{C}}$

is equal to $\cfrac{A}{1}$*$\cfrac{C}{B}=\cfrac{AC}{B}$

The second fraction you put up is incorrect.

$\cfrac{B}{\frac{C}{A}}$

is equal to $\cfrac{B}{1}*\cfrac{A}{C}=\cfrac{BA}{C}$

Think of division $\cfrac{A}{B}$ as multiplication of $A*\cfrac{1}{B}$

answered Jan 29 '18 at 02:07

Yash Jain

1

I use symbolab to check my work -- are they an approved tool? They say they used the incorrect one as part of a soln -- maybe the translation from latex to plain text is just wrong on their end? Also, you can use cfrac for nicer fractions. Thanks for your help. – user2355058 Jan 29 '18 at 02:11
1

All online calculators should be correct as long as you are specifying the parenthesis. For instance, $A/BC$ is not the same as $A/(BC)$ – Yash Jain Jan 29 '18 at 02:13
Ross Millikan's answer is far superior, @user2355058. – Shaun Jan 29 '18 at 02:38
1

@Shaun which makes it a reason for you to downvote my corrct answer which i posted before the other person – Yash Jain Jan 29 '18 at 02:50

Fraction rules A/B/C vs B/C/A

2 Answers2