# How to simplify compounded division

How would you simplify something like: $$(1184 / 60) / 60$$ How do you figure out what to divide the number by just once to come out the same as when you divide it twice?

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$(a/b)/c=a/(bc)$.${}{}{}{}{}{}$

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Perfect, thanks. what do you call this procedure? if anything. –  Marshall House May 29 '12 at 3:28
I call it "understanding how division works," but I don't think it has any snappy official-sounding name. –  Gerry Myerson May 29 '12 at 3:30
lol long day. Normally i would think about it but i've been meaning to start participating on here anyhow. –  Marshall House May 29 '12 at 3:34
@Marshall: You could call it "multiplying numerator and denominator by $b$" as in $$\frac{\frac{a}{b}}{c}=\frac{b\times\frac{a}{b}}{b\times c} =\frac{a}{b c}$$ –  Henry May 29 '12 at 6:56

$\rm Hint\!:\ (A/B)/C\ = X$

$\rm\ \ \iff\ A/B\ =\ C\ X$

$\rm\ \ \iff\ \ \ \: A\, =\, BC\ X$

$\rm\ \ \iff\ A/(BC) = X$

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Sandwich: The bread goes on top, the ham and cheese in the bottom: $$\frac{\quad\frac{a}{b}\quad{\leftarrow\text{bread}\atop \leftarrow\text{cheese}}\quad}{\quad\frac{c}{d}\quad{\leftarrow\text{ham}\atop\leftarrow\text{bread}}\quad} = \frac{ad}{bc}\quad{\leftarrow\text{bread}\atop\leftarrow\text{ham and cheese}}$$

You have $a=1186$, $b=60$, $c=60$, and $d=1$: $$\frac{\quad\frac{1186}{60}\quad}{60} = \frac{\quad\frac{1186}{60}\quad}{\frac{60}{1}} = \frac{(1186)(1)}{(60)(60)}.$$

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Oi va voi! Such treyf! Readers whose sensibilities are offended can substitute lox and cream cheese. –  Gerry Myerson May 29 '12 at 3:32
why is "d" necessary? –  Marshall House May 29 '12 at 3:35
@Marshall: It's not strictly necessary, but with it, your problem fits into the general scheme of dividing a fraction by a fraction. Likewise, if you had $$\frac{a}{\quad\frac{c}{d}\quad}$$you could just try to remember that it's the same as $\frac{ad}{c}$, or you could make it fit into the general scheme of dividing a fraction by a fraction. Since dividing a fraction by a fraction shows up a lot, might as well see the general case. –  Arturo Magidin May 29 '12 at 3:37
Now you made me hungry. –  Pedro Tamaroff May 29 '12 at 5:00