Triple fractions

Question

I've got this simple assignment, to find out the density for a give sphere with a radius = 2cm and the mass 296g. It seems straightforward, but it all got hairy when i've got to a fraction with three elements(more precisely a fraction divided by a number actually this was wrong, the whole point was that the number is divided by a fraction, and it's different than a fraction being divided by a number.). I tend to solve these by dividing the element on the bottom by 1, and extracting from that 2 fraction division like this :

$$ \frac{a}{\frac{b}{c}} \Rightarrow \frac{\frac{a}{b}}{\frac{c}{1}} \Rightarrow \frac{a}{b} \div \frac{c}{1} => \frac{a}{b} \cdot \frac{1}{c} \Rightarrow \frac {a} {b \cdot c} $$

And it used to work, though for the next example it doesn't seem to, it looks like another technique is used:

$$ \frac{a}{\frac{b}{c}} \Rightarrow a \div \frac{b}{c} \Rightarrow a \cdot \frac{c}{b} \Rightarrow \frac {a \cdot c}{ b} $$

For the example below cleary the second method is used/needed, to get the right response. But i'm confused when to use each, as i've use both before, and both gave correct asnwers(matching with the answers at the end of the book).

$$ v = \frac43\pi r^3 $$

$$ d = \frac mv $$

$$ m = 296g $$

$$ r=2cm $$

$$ v = \frac43\pi 2^3 \Rightarrow \frac{32\pi}{3} $$

$$ d = \frac{m}{v} \Rightarrow \frac{296}{\frac{32\pi}{3}} \Rightarrow \frac {296}{32\pi} \div \frac31 \Rightarrow \frac{296}{32\pi} \cdot \frac{1}{3} \Rightarrow \frac{296}{96\pi} \approx 0.9814\frac{g}{cm^3} $$

$$ d_{expected} = 8.8 \frac{g}{cm^3} $$

I am, clearly, missing something fundamental about the use of these.

Can anyone enlighten me please? Can't quite find a good explanation online.

Answer 1

$$\frac{a}{\frac{b}{c}}\ne\frac{\frac{a}{b}}{c} \tag 1$$

The left-hand side of $(1)$ can be written as

$$\frac{a}{\frac{b}{c}}=\frac{ac}{b}$$

whereas the right-hand side of $(1)$ can be written as

$$\frac{\frac{a}{b}}{c}=\frac{a}{bc}$$

Let's look at an example: Suppose $a=3$, $b=6$, and $c=2$. Then, we have

$$\frac{a}{\frac{b}{c}}=\frac{3}{\frac{6}{2}}=\frac{3}{3}=1$$

but

$$\frac{\frac{a}{b}}{c}=\frac{\frac{3}{6}}{2}=\frac{1/2}{2}=\frac{1}{4}$$

Answer 2

In $\frac{a/b}{c}=\frac{a}{bc}$, we divide $a$ by $b$, then divide the result by $c$.

In $\frac{a}{b/c}=\frac{ac}{b}$, we divide $b$ by $c$, then divide $a$ by the result.

The two mean different things.

Answer 3

The reason you're getting non-equivalent expressions from $\frac{a}{\frac{b}{c}}$ is because of the step $\frac{a}{\frac{b}{c}} \Rightarrow \frac{\frac{a}{b}}{\frac{c}{1}}$. It should be $\frac{a}{\frac{b}{c}} \Rightarrow \frac{\frac{a}{b}}{\frac{1}{c}}$

Answer 4

296/(32pi/3)=(296/32pi)/(3/1) should be =(296/32pi)*(3/1)= 8.833 g/ccm you inverted twice

Answer 5

to use your "introduce 1 to the denominator" approach, you should instead put that 1 with $a$:

$\frac{a}{\frac{b}{c}}=\frac{\frac{a}{1}}{\frac{b}{c}}=\frac{\frac{ac}{c}}{\frac{b}{c}}=\frac{ac}{b}$

(for the same reasons that the others point out)