# Division - with Division in the Denominator

#### rickS

Would anyone be so kind as to show me how to simplify (reduce) this equation? (Happy)

#### Shakarri

$$\displaystyle x = \frac{n}{\frac{nx}{b}}$$

First simplify the right hand side

When you are dividing by a fraction you flip the denominator around and multiply by it.

So $$\displaystyle \frac{n}{\frac{nx}{b}}$$ becomes $$\displaystyle n \times \frac{b}{nx}$$

$$\displaystyle n \times \frac{b}{nx} = \frac{nb}{nx}$$

n is in the numerator and the denominator so they cancel each other out

$$\displaystyle \frac{nb}{nx} = \frac{b}{x}$$

Putting the simplified right hand side back into the original equation gives:

$$\displaystyle x = \frac{b}{x}$$

I think you can take it from here

rickS

#### rickS

Shakarri, thanks!

Just to clarify...simplifying the right side happens without touching "x" on the left side of the equation? (i.e. without flipping the denominator around and multiply by it on the left side of the equation also.

#### Shakarri

That's correct, we took the right hand side alone to simplify the fraction

#### JeffM

Shakarri, thanks!

Just to clarify...simplifying the right side happens without touching "x" on the left side of the equation? (i.e. without flipping the denominator around and multiply by it on the left side of the equation also.
If we did anything that changed the value of the right hand side, we would have to change the left hand side correspondingly. But we are not changing the value of the right hand side, and so no change is needed on the left hand side.

$x = \dfrac{n}{\dfrac{nx}{b}} = \dfrac{\cancel n}{1} * \dfrac{b}{\cancel nx} = \dfrac{b}{x}.$

rickS

#### rickS

Jeff and Shakarri ... thanks for clarifying.

I am so impressed with the talent and ethos of this forum ... thanks for being a part of that!

#### Archie

Another way to think of this is that you multiply both the numerator and the denominator of the RHS by the same value $$\displaystyle \left({b \over nx}\right)$$ which makes the denominator equal to 1, so it can be discarded and we are left only with the numerator (which is now a fraction).
$$\displaystyle {n \over {nx \over b}} = {n \times {b \over nx} \over {nx \over b} \times {b \over nx}} = {{nb \over nx} \over 1} = {nb \over nx} = {b \over x}$$

rickS

#### rickS

Archie, awesome explanation!