The ratio of alcohol to water in two containers, A and B, is 5:3 and 1:3, respectively, with both containers having infinite capacity. Suppose that the aim is to obtain 2.1 litres of liquid, which is composed of equal quantities of alcohol and water. How much liquid should be drawn from A (in litres)?
Step-by-step Solution:
The final mixture must have equal quantities of alcohol and water (1:1 ratio). This means the total alcohol needed is half of 2.1 litres: $$ \text{Total Alcohol} = \frac{1}{2} \times 2.1 = 1.05 \text{ litres} $$ Let the amount drawn from container A be $x$ litres, and from container B be $y$ litres. $$ x + y = 2.1 \implies y = 2.1 - x $$ The proportion of alcohol in A is $\frac{5}{8}$. The proportion of alcohol in B is $\frac{1}{4}$. The total alcohol equation becomes: $$ \frac{5}{8}x + \frac{1}{4}(2.1 - x) = 1.05 $$ $$ \frac{5}{8}x + \frac{0.525}{x} - \frac{2}{8}x = 1.05 $$ $$ \frac{3}{8}x + 0.525 = 1.05 $$ $$ \frac{3}{8}x = 0.525 $$ $$ x = 0.525 \times \frac{8}{3} $$ $$ x = 0.175 \times 8 = 1.4 \text{ litres} $$