How to determine mixing ratio to get desired percentage?

Let's say I have two types of chocolate. Chocolate A contains 70% cocoa and 30% sugar. Chocolate B contains 100% cocoa and no sugar.

How can I figure out the exact mixing ratio I need to use to combine these two chocolates into a new product that contains 90% cocoa and 10% sugar?

As one of our constraints we could use the fact that $$x$$ parts of chocolate $$A$$ plus $$y$$ parts of chocolate $$B$$ should add up to a whole. Furthermore, if a certain amount of chocolate $$A$$ contains $$70\%$$ cocoa, then exactly the same amount of chocolate $$B$$ will contain $$100\%$$ cocoa. Therefore, $$70\%$$ of the $$x$$ parts plus $$100\%$$ of the $$y$$ parts should give us $$90\%$$ of the entire whole. With these constrains in place, we get the following system of two equations with two unknowns:

$$\begin{cases} x+y=1\\ 0.7x+y=0.9. \end{cases}$$

Solving this system of equations, we get:

$$0.7x+(1-x)=0.9\implies x=\frac{1}{3},\\ y=1-x\implies y=\frac{2}{3}.$$

So, the sought ratio then will be (multiply the whole thing through by $$3$$ to get rid of those fractions):

$$\frac{1}{3}:\frac{2}{3}\implies 3\cdot\frac{1}{3}:3\cdot\frac{2}{3}\implies 1:2.$$

Thus, the mixing ratio you need to use to combine these two chocolates into a new product that contains $$90\%$$ cocoa and $$10\%$$ sugar is one part of chocolate $$A$$ and two parts of chocolate $$B$$.

Suppose we have $$x$$ parts of chocolate A. Then we have 0.7x cocoa and 0.3x sugar, but to get 90% cocoa we need 2.7x cocoa. This obviously means that we need 2x parts of chocolate B, so the answer is that the A:B ratio is $$1:2$$

• Thanks for the reply! Is there a formula or equation I could use to calculate the same question, but with other numeric values as well? – Emiliy Jun 8 at 23:49
• Yes, there is. Suppose Chocolate A has $a$ cocoa and $1-a$ sugar, and Chocolate B has $b$ cocoa and $1-b$ sugar (where $0\le a\le 1$ and $0\le b\le 1$), and your target is to get $c$ cocoa. (Note that your target is attainable from mixing Chocolate A and B if and only if $c$ is between $a$ and $b$.) Suppose you need to mix Chocolate A to Chocolate B in ratio $r:(1-r)$ for some $r\in [0,1]$ (your goal is to find $r$ in terms of $a,b,c$). Then the mixture has the following amount of cocoa: $$ra + (1-r)b.$$ Since we want this to equal $c$, we need $$\boxed{r = \frac{c-b}{a-b}},$$ for $a\ne b$. – Minus One-Twelfth Jun 9 at 3:25
• For example, in your OP, we have $a=0.7, b=1, c=0.9$. Using the above formula, $r=\frac{0.9-1}{0.7-1}=\frac{1}{3}$. Hence we mix Chocolate A and Chocolate B in the ratio $r:(1-r)=\frac{1}{3}:\frac{2}{3}=1:2$, agreeing with the answers above. – Minus One-Twelfth Jun 9 at 3:27