The ratio of alcohol to water in two containers, $A$ and $B$, are $5:3$ and $1:3$, respectively, with both containers having infinite capacity. Suppose that the aim is to obtain $2.1$ litre of liquid which is composed of equal quantities of alcohol and water. How much liquid should be drawn from $A$?

We have two containers:

Container $A$: alcohol : water $= 5:3 \implies$ alcohol fraction $= \dfrac{5}{8}$

Container $B$: alcohol : water $= 1:3 \implies$ alcohol fraction $= \dfrac{1}{4}$

Target: $2.1$ litres with alcohol $=$ water $= 1.05$ litres each.

Let $a =$ litres drawn from $A$, and $b =$ litres drawn from $B$.

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$a + b = 2.1 \quad \text{...(1)}$

$\frac{5a}{8} + \frac{b}{4} = 1.05 \quad \text{...(2)}$

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Elimination:

Multiply $(2)$ by $8$:

$5a + 2b = 8.4 \quad \text{...(3)}$

Multiply $(1)$ by $2$:

$2a + 2b = 4.2 \quad \text{...(4)}$

Subtract $(4)$ from $(3)$:

$3a = 4.2 \implies a = 1.4$

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Finding $b$:

Substitute $a = 1.4$ into $(1)$:

$1.4 + b = 2.1 \implies b = 0.7$

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Draw $1.4$ litres from $A$ and $0.7$ litres from $B$.