How do I calculate combining two ratios to create a new third ratio

Question

Can someone provide me with a formula that directs me in combining two known ratios to create a new third ratio?

Solution 1 contains 43mg ingredient A, and 3.7mg ingredient B for a ratio of about 12:1

Solution 2 contains 0.69mg ingredient A, and 48mg ingredient B for a ratio of about 1:70

Is there a formula I can use directing how much of each Solution to use to create a new solution with a ratio about 1:1?

Answer 1

Compute the amount of each ingredient. We might as well start with $46.7$ mg of solution $1$ and add $x$ mg of solution $2$. Then we have $43+\frac {0.69}{48.69}x$ of $A$ and $3.7+\frac {48}{48.69}x$ of $B$. Equate those and solve for $x$.

Answer 2

The first solution has $43A+3.7B$ and the second has $0.69A+48B$ we want a linear combination of these that the same amount of $A$ and $B$. \begin{eqnarray*} \lambda(43A+3.7B)+\mu(0.69A+48B)=(43 \lambda+0.69 \mu)A+(3.7 \lambda+48 \mu)B \end{eqnarray*} So we need \begin{eqnarray*} 43 \lambda+0.69 \mu&=&3.7 \lambda+48 \mu \\ 39.3 \lambda&=&47.31 \mu \\ \end{eqnarray*} and $\lambda=47.31,\mu=39.3$ is an obvious solution.

Answer 3

Let the amount of Solution $1$ be $\alpha$ and of Solution $2$ be $\beta$. Then: $$43\alpha+0.69\beta=3.7\alpha+48\beta$$ This leads to: $$39.3\alpha=47.31\beta\to\alpha=\frac{4731}{3930}\beta$$ So use approximately $1.2$ units of Solution $1$ per unit of Solution $2$