Is either $\dfrac{A_1}{B_1}=.25$, $\dfrac{A_2}{B_2}=.4$ or $\dfrac{A_1}{B_1} +\dfrac{A_2}{B_2}=.35$?

Question

Solution $A$, containing $25 \%$ oil and $75 \%$ water, is mixed with Solution $B$, containing $40 \%$ oil and $60 \%$ water. If the resulting mixture contains $35 \%$ oil, approximately what proportion of the total mixture is made up by Solution $A$ ?

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My work:

Let $A_1$ the proportion of the total mixture that is oil for solution A. Let $B_1$ the volume of the solution A.

Let $A_2$ the proportion of the total mixture that is oil for solution B. Let $B_2$ the volume of the solution B.

I think $\dfrac{A_1}{B_1}=.25$, $\dfrac{A_2}{B_2}=.4$ $\iff$ this is a contradiction with the resulting mixture being $35\%$ oil

I also think $\dfrac{A_1}{B_1} +\dfrac{A_2}{B_2}=.35$

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The question asks to solve for $\dfrac{A_1}{B_1+B_2}$

Answer 1

You are close! But, you don't necessarily know that $A$ and $B$ are in equal proportion - in fact they are not.

Let $x$ be the proportion of the final mixture which is from $A$, and let $(1-x)$ be the proportion from $B$. Then $25x + 40(1-x) = 35$. So, we simply find that we have a linear equation, and we solve for $x = 1/3$. That is, the ratio of $A$ to $B$ in the final mixture is 1:2.

Hope this helps!

Answer 2

Solution $A$ has a volume (or mass) $V_A$, and the quantity of oil in that is $0.25 V_A$. The second solution has a volume $V_B$ and the amount of oil is $0.4V_B$. Then $$\frac{0.25 V_A+0.4V_B}{V_A+V_B}=0.35$$Divide both numerator and denominator on the left hand side by $V_B$: $$\frac{0.25 V_A/V_B+0.4}{V_A/V_B+1}=0.35$$You can now simply calculate what is the $V_A/V_B$ fraction. Let's call it $v$. Then the question is looking for $$\frac{V_A}{V_A+V_B}=\frac{V_A/V_B}{V_A/V_B+1}=\frac v{v+1}$$

Answer 3

Your notation is rather confusing, so I will use $V_A,V_B$ for the volume of the two solutions. Thus, the volume of oil is $0.25V_A+0.4V_B$. We are given that in the final solution, oil makes up $35\%$ of it's volume. So, $$\frac{0.25V_A+0.4V_B}{V_A+V_B}=0.35$$

$p= \frac{V_A}{V_A+V_B}$ is the proportion of solution $A$ in the mixture, and we can now solve for the answer.

$$0.25 p+0.4(1-p)=0.35$$ $$\implies p=\frac13$$