Mixtures & Alligation

At a glance 10 formulas

1. 1Mean of a mixture$\text{mean} = \dfrac{q_1 x_1 + q_2 x_2}{q_1 + q_2}$
2. 2Alligation cross$\dfrac{q_{\text{cheap}}}{q_{\text{dear}}} = \dfrac{d - m}{m - c}$$\text{the mean } m \text{ lies between } c \text{ and } d$
3. 3Balance the deviations$\textstyle\sum q_i\,(x_i - m) = 0$$\text{three or more items mixed to a mean } m$
4. 4Topping up one part$\text{new total} = \dfrac{\text{fixed part}}{\text{its new share}}$$\text{to add} = \text{new total} - \text{old total}$
5. 5Strength p% to t%$\dfrac{\text{mix}}{\text{pure}} = \dfrac{100 - t}{t - p}$$\text{water: } \dfrac{\text{mix}}{\text{water}} = \dfrac{t}{p - t}$
6. 6Drawing off a mixtureMixture drawn off leaves in its current ratio, so take each part out in proportion, then add what is poured in.
7. 7Repeated replacement$\text{left} = V\left(1 - \dfrac{x}{V}\right)^{n}$$\text{strength: } c_n = c_0\left(1 - \dfrac{x}{V}\right)^{n}$
8. 8Cost before profit$\text{mean cost} = \dfrac{100 \times SP}{100 + g}$
9. 9Water costs nothing$\dfrac{\text{water}}{\text{milk}} = \dfrac{g}{100}$$\text{mix sold at milk's cost price, gain } g\%$
10. 10Alloys and ratiosTurn each ratio into the share of one metal (5 : 3 is $\tfrac{5}{8}$ copper), then alligate the shares, never the ratios.