Determining ratio of a 2 part mix

[rustyskills]

Mechanically reconditioned sand weighs 8.50#/gallon
Thermally reclaimed sand weighs 8.89#/gallon
A mix of these two kinds of sand weighs 8.78#/gallon
What is the % thermal and %mechanical in the mix?

I work in a foundry and we feed these two types of sand into a hopper that feeds a mixer. It's important to have a quick and reliable test that can be done on the shop floor that confirms that the mix is 60/40 or 70/30 etc.

I looked at a lot of calculators and examples online but didn't find any that fit the data I have collected. I sure would appreciate some help. Anyone want to set me straight on this? Thx

 

[MarkFL]

Let's let $M$ be the percentage of mechanically reconditioned sand and $T$ be the percentage of thermally reconditioned sand.

So, we know that:

\(\displaystyle M+T=100\)

And we also know:

\(\displaystyle 850M+889T=87800\)

Solving this system, we find:

\(\displaystyle M=\frac{1100}{39}\approx28.2,\,T=\frac{2800}{39}\approx71.8\)

You could also state:

\(\displaystyle M:T=11:28\)

 

[rustyskills]

modern day (math) warrior, mean mean stride...
thank you. I can duplicate that but I don't fully understand it yet.
I get the 850M + 889T = 87800
can you show me a little more about how you got to the rest?
much appreciated.

 

[MarkFL]

Suppose we take the first equation:

\(\displaystyle M+T=100\)

and multiply it by $-850$, so that it becomes:

\(\displaystyle -850M-850T=-85000\)

Now, recall we have:

\(\displaystyle 850M+889T=87800\)

Adding these two equation, we will eliminate $M$, to get:

\(\displaystyle 39T=2800\)

which gives us:

\(\displaystyle T=\frac{2800}{39}\)

And then taking the original first equation, and using this value for $T$, we have:

\(\displaystyle M=100-\frac{2800}{39}=\frac{1100}{39}\)

 

[rustyskills]

There it is! I guess this answers the age old question "Why do I have to learn this- I'll never use it." Math in the foundry... I thank you.
I still have some R&R concerns regarding sampling technique but we can easily test and measure that. The roadblock was the calculation of the mix and you have cleared that right up.