# I need an equation: I have the weights of two ingredients...

#### 16_hope_dylan

##### New member
I have the weights of two ingredients (Weight of ingredient 1 = r. Weight of ingredient 2 = i) and the two ingredients need to fit into a volume(volume in cm cubed = v) and the two ingredients also need to be in a variable ratio (r/k = i/s, where k and s can be any two numbers that have a sum of 100). Also, the density of ingredient 1 is 2.1grams/1cm cubed and the density of ingredient 2 is 0.8grams/1cm cubed. What equation(s) can I use to find r and i if I input the other variables? Is such an equation even possible?

#### Subhotosh Khan

##### Super Moderator
Staff member
I have the weights of two ingredients (Weight of ingredient 1 = r. Weight of ingredient 2 = i) and the two ingredients need to fit into a volume(volume in cm cubed = v) and the two ingredients also need to be in a variable ratio (r/k = i/s, where k and s can be any two numbers that have a sum of 100). Also, the density of ingredient 1 is 2.1grams/1cm cubed and the density of ingredient 2 is 0.8grams/1cm cubed. What equation(s) can I use to find r and i if I input the other variables? Is such an equation even possible?
What are "r" & "i"? Please describe.

#### 16_hope_dylan

##### New member
“r” and “i” are the output values. “r” is the weight of one ingredient(ingredient 1) in a mixture and “i” is the weight of another ingredient(ingredient 2) in the mixture

#### HallsofIvy

##### Elite Member
I have the weights of two ingredients (Weight of ingredient 1 = r. Weight of ingredient 2 = i) and the two ingredients need to fit into a volume(volume in cm cubed = v) and the two ingredients also need to be in a variable ratio (r/k = i/s, where k and s can be any two numbers that have a sum of 100). Also, the density of ingredient 1 is 2.1grams/1cm cubed and the density of ingredient 2 is 0.8grams/1cm cubed. What equation(s) can I use to find r and i if I input the other variables? Is such an equation even possible?
If you have weight r of ingredient 1 (I presume in grams) and its density is 2.1 grams per cubic cm, then its volume is r/2.1 cubic cm. Similarly if the weight of ingredient 2 is i grams and its density is 0.8 grams per cubic cm then its volume is i/0.8 cubic cm. You want that to fit into volume v cubic cm. so r/2.1+ i/0.8= v. I don't like fractions so I would multiply both sides of the equation by 2.1(0.8)= 1.68. The equation becomes 0.8r+ 2.1i= 1.68v.

Now you say "the two ingredients also need to be in a variable ratio (r/k = i/s, where k and s can be any two numbers that have a sum of 100)". Leaving k as a parameter, s= 100- k so that second equation becomes r/k= i/(100- k). Multiplying both sides by 100- k, i= [(100- k)/k]r= (100/k- 1)r= (100/k)r- r. So now the first equation becomes 0.8r+ 2.1[(100/k)r- r= 0.8r- 2.1r+ (210/k)r= (210/k- 1.3)r= 1.68v. r= 1.68v/(210/k- 1.3)= 1.68kv/(210- 1.3k).

Once you have that value of r, i= (1.68v- 0.8r)/2.1. Of course both r and i will depend on the two parameters, v and k.

#### 16_hope_dylan

##### New member
If you have weight r of ingredient 1 (I presume in grams) and its density is 2.1 grams per cubic cm, then its volume is r/2.1 cubic cm. Similarly if the weight of ingredient 2 is i grams and its density is 0.8 grams per cubic cm then its volume is i/0.8 cubic cm. You want that to fit into volume v cubic cm. so r/2.1+ i/0.8= v. I don't like fractions so I would multiply both sides of the equation by 2.1(0.8)= 1.68. The equation becomes 0.8r+ 2.1i= 1.68v.

Now you say "the two ingredients also need to be in a variable ratio (r/k = i/s, where k and s can be any two numbers that have a sum of 100)". Leaving k as a parameter, s= 100- k so that second equation becomes r/k= i/(100- k). Multiplying both sides by 100- k, i= [(100- k)/k]r= (100/k- 1)r= (100/k)r- r. So now the first equation becomes 0.8r+ 2.1[(100/k)r- r= 0.8r- 2.1r+ (210/k)r= (210/k- 1.3)r= 1.68v. r= 1.68v/(210/k- 1.3)= 1.68kv/(210- 1.3k).

Once you have that value of r, i= (1.68v- 0.8r)/2.1. Of course both r and i will depend on the two parameters, v and k.
So I kinda get your work but towards the ending it gets a little confusing. What of those equations should I be using?