Percentages and fractions - turning them upside down?

[Spaceghost1]

Homework Statement

Hello, I have a workbook here with a question I am a little confused about, I think it should be quite simple to answer. I know the answer but I don't really understand how it is applied. I know the method but I want clarification on how it works please.

The average abundance of gold in the Earth's crust is 4.0 x 10^-7%. This means that there is 4.0 x 10^-7
kg of gold in 100 kg of the the crust or:

4.0 x 10^-7 kg
_____________ as a fraction.
100 kg

Now this is where I get confused.

Doing the fraction

100 kg
__________________
4.0 x 10^-7 kg

tells me that there is 1 kg of gold in 2.5 x 10⁸kg of crust. So basically the fraction has just been turned upside down (is there a proper term for doing this?) but why does the answer tell me how much crust comes from 1 kg of gold? Where did 1 come from?

Thanks.

 

[Mentallic]

It's called taking the reciprocal and it works like this:

If we have the gold:crust ratio (or fraction) as being x:100 (in this case, 4.0 x 10^-7 : 100) then it is also the same to say that we have x/100 : 1 (we divided both sides by 100, if you prefer fractions it works just the same).

$$\frac{4.0\cdot 10^{-7}}{100}=\frac{(4.0\cdot 10^{-7})/100}{100/100}=\frac{4.0\cdot 10^{-9}}{1}$$

Ok now so we have 4.0 x 10^-9 kg of gold for every kg of crust. But what if we wanted to know how much crust we have for every kg of gold?

All we do is solve this equation:

$$\frac{4.0\cdot 10^{-9}}{1}=\frac{1}{x}$$

Where x is the amount of crust. This equation in words says there is a tiny amount of gold in every unit of crust, which is equivalent to a unit of gold in a large portion of crust. All you have to do is solve for x now and you'll see that you've taken the reciprocal of the fraction.

 

[Spaceghost1]

Thank for your help, I understand it now. For the last equation I think you meant to say x is the amount of gold rather than crust?

 

[Yayness]

There is 4.0·10^-9 kg of gold in every kg of crust.
This is the same as saying that one kg of crust contains 4.0·10^-9 kg of gold.

That means 2 kg of crust contains 2·4.0·10^-9 kg of gold, and 3 kg of crust contains 3·4.0·10^-9 kg of gold.
If we let x be how many kg of crust there is, then: x kg of crust contains x·4.0·10^-9 kg of gold.

Lets say you want to know how much crust there would be if there were 1 kg of gold inside.
We know that x kg of crust contains x·4.0·10^-9 kg of gold, and we want there to be 1 kg of gold.
This means that: x·4.0·10^-9 = 1
Divide both sides by 4.0·10^-9 and we get:
$$x=\frac{1}{4.0\cdot 10^{-9}}$$
And as I said above, x is how many kg of crust there is.

 

[Mentallic]

Thank for your help, I understand it now. For the last equation I think you meant to say x is the amount of gold rather than crust?

No I still stand by what I said. In that equation, we are trying to find the amount of crust (given the variable x) such that there is 1kg of gold in it, given that there is 4.0$\cdot$10^-9kg of gold in 1kg of crust. Algebraically and maybe even intuitively you can show that $$x=\frac{1}{4.0\cdot 10^{-9}}=2.5\cdot 10^8$$.

You can think about it a little more to get a better understanding of it using different numbers. If we have half of the 1kg crust made out of gold, then we have 1kg of gold in 1/half or 2kg of crust.
0.1kg of gold in 1kg of crust gives 1/0.1=10kg of crust with 1kg of gold.

etc.