Mixtures & Alligation

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Mixture Concentration & Amounts

Amount from ratio

\text{part} = \text{total} \times \frac{\text{part's share}}{\text{sum of shares}}

Concentration

C = \frac{\text{ingredient}}{\text{mixture}} \times 100\%

Adding water

C' = \frac{A}{M + w} \quad (A \text{ unchanged})

Removing mixture

\text{each ingredient} \to \text{same fraction removed}

Alligation ratio

\frac{n_1}{n_2} = \frac{v_2 - \bar{v}}{\bar{v} - v_1}

Weighted average

\bar{v} = \frac{n_1 v_1 + n_2 v_2}{n_1 + n_2}

Cheaper : dearer

(d - m) : (m - c)

Water as free ingredient

\text{milk} : \text{water} = (m - 0) : (c - m), \ c > m

Repeated equal replacement

C\left(1 - \frac{r}{C}\right)^n

Milk : water after n ops

\left(1-\frac{r}{C}\right)^n : \left[1-\left(1-\frac{r}{C}\right)^n\right]

Proportional removal

\text{lost} = \text{removed volume} \times \frac{\text{ingredient share}}{\text{total}}

Chain of replacements

C \prod_k \left(1 - \frac{r_k}{C}\right)

Adulteration profit

\text{profit\%} = \frac{\text{water}}{\text{milk}} \times 100

Ratio target

\frac{m}{w + x} = \text{wanted ratio} \Rightarrow x

Alloy rebuild

\text{metal} = \text{alloy weight} \times \frac{\text{share}}{\text{sum of shares}}

SP of mixture

\text{SP} = (\text{milk}) \times \text{milk price} + (\text{water}) \times 0

Blend of two mixtures

f = \frac{V_1 f_1 + V_2 f_2}{V_1 + V_2}

Volumes for target fraction

\frac{V_1}{V_2} = \frac{f_2 - f}{f - f_1}

Fraction from ratio

f_{\text{milk}} = \frac{m}{m + w}