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

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?

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 $x$.

• If you take $x$ mg of solution $1$ and $y$ mg of solution $2$, then we'll get simultaneous equations to solve. Am I right ?? @RossMillikan Commented Aug 5, 2018 at 19:05
• Yes. You could then set $x+y$ to the total amount of solution you want at the end. With my approach you just have one equation to solve, then need to scale up or down to get the total amount desired. Commented Aug 5, 2018 at 19:07
• Got it !! That's why you fixed $x$, I see !! Commented Aug 5, 2018 at 19:08

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.

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 $2$

• Rhys, your equation is easy for me to understand and solve, thank you! I theorized that if instead of a 1:1 solution I was trying to create a new solution with a 2:1 or 3:1 ratio how your equation would need to be changed to accommodate. I thought it would be as simple as multiplying each side of the equation by the appropriate ratio value but when I did the math the result was obviously wrong. Here's what I thought might work to calculate a 2:1 solution... 2(43α + 0.69ß) = 1(3.7α + 48ß) Solving for α= 0.566, seems like the correct answer should be 2x the result that you got of 1.23?
– Tom
Commented Aug 9, 2018 at 1:19
• If you want twice as much of ingredient $A$ as ingredient $B$, then the first equation becomes $$43\alpha+0.69\beta=2(3.7\alpha+48\beta)$$. Then we get $$35.6\alpha=95.31\beta\to \alpha=\frac{9531}{3560}\beta$$. This comes out as around 2.68 units of Solution 1 per unit of Solution 2. Commented Aug 10, 2018 at 10:41
• Thanks Rhys for your help, I really appreciate it!
– Tom
Commented Aug 12, 2018 at 15:41