Mixtures & Alligation
🔒 Log in to trackhigh importance~1 Q in Tier 120 formulas⚡ 15 shortcuts5 subtopics
Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.
Mixture Concentration & Amounts
Amount from ratio
\text{part} = \text{total} \times \frac{\text{share}}{\text{sum of shares}}
Concentration
C = \frac{\text{ingredient}}{\text{mixture}} \times 100\%
Mean price
\text{mean} = \frac{q_1 p_1 + q_2 p_2}{q_1 + q_2}
Adding water
C' = \frac{A}{M + w}
A = amount of the other ingredient, unchanged.
Removing mixture
\text{each ingredient shrinks in its own share}
A uniform draw keeps the ratio.
Rule of Alligation
Alligation ratio
\frac{\text{cheaper}}{\text{dearer}} = \frac{D - M}{M - C}
Weighted average
\bar{v} = \frac{n_1 v_1 + n_2 v_2}{n_1 + n_2}
Water as the free ingredient
\text{milk} : \text{water} = (m - 0) : (c - m),\ c > m
Replacement & Repeated Operations
Repeated equal replacement
\text{left} = C\left(1 - \frac{x}{C}\right)^n
Milk : water after n rounds
\left(1-\frac{x}{C}\right)^n : \left[1-\left(1-\frac{x}{C}\right)^n\right]
Proportional removal
\text{lost} = \text{drawn volume} \times \frac{\text{ingredient share}}{\text{total}}
Unequal draws
C \prod_k \left(1 - \frac{x_k}{C}\right)
Multiply one factor per round when the draws differ.
Milk–Water Ratio & Profit by Adulteration
Adulteration gain
\text{gain\%} = \frac{\text{water}}{\text{milk}} \times 100
Target ratio
W : M = g : 100
For a gain of g% when sold at cost price.
Blend cost price
CP = \frac{q_1 c_1 + q_2 c_2}{q_1 + q_2}, \quad SP = CP\left(1 + \frac{g}{100}\right)
Alloy rebuild
\text{metal} = \text{alloy weight} \times \frac{\text{share}}{\text{sum of shares}}
Mixing Two Mixtures
Blend of two mixtures
f = \frac{V_1 f_1 + V_2 f_2}{V_1 + V_2}
Volumes for a target fraction
\frac{V_1}{V_2} = \frac{f_2 - f}{f - f_1}
Fraction from ratio
f_{\text{milk}} = \frac{m}{m + w}
Equal volumes
\bar{f} = \frac{f_1 + f_2 + f_3}{3}
Only when every vessel holds the same volume.