Mixtures & Alligation

Formula sheet

Everything for this chapter, in one place.

The one rule


Always track	the amount of each component, never the percentage
Amount	$\text{fraction} \times \text{total volume}$
Why	percentages shift when the total changes; amounts only change when you add or remove

Alligation — mixing two things of known strength


Quantities $x$ at strength $p$ , $y$ at strength $q$ , result $r$	$\dfrac{x}{y} = \dfrac{q-r}{r-p}$
In words	the distances from the mean are in the reverse ratio of the quantities
It works for	concentration, price per kg, density, or any other "per unit" quantity
It does not work for	anything that is not per-unit — totals, or quantities themselves

Successive replacement — remove some, top up, repeat


Remove fraction $f$ each time, $n$ times	remaining original $=$ start $\times (1-f)^n$
Removing volume $v$ from total $V$	remaining original $=$ start $\times \left(1-\dfrac{v}{V}\right)^n$
The total volume	never changes
Ratio after $n$ rounds	$(1-f)^n : 1-(1-f)^n$

Transfers between two vessels


Take $v$ from A to B, stir, take $v$ back	the amount of A's liquid now in B equals the amount of B's liquid now in A
Why	both vessels end with the volume they began with, so whatever left must have been replaced

Mixing by weight and by volume


Volumes in ratio $a:b$ , densities in ratio $c:d$	weights are in ratio $ac : bd$
Strength $p\%$	$p$ grams in every 100 ml

With cost and profit


Cost of the mixture per unit	the weighted average of the component costs
Selling price	$\text{cost} \times (1 + \text{profit }\%)$
Part sold at a different price	work in totals — total revenue needed minus revenue already taken