Two liquids A and B are in the ratio 5:1 in container 1 and in the ratio 1:3 in container 2. In what ratio should the contents of the two containers be mixed so as to obtain a mixture of A and B in the ratio 1:1?
Let the required ratio be \[ x:y \]. Given Information: - Container 1 (Mixture Composition): - Liquid A in \( x \) litres of mixture: \[ \frac{5x}{6} \text{ litres} \] - Liquid B in \( x \) litres of mixture: \[ \frac{x}{6} \text{ litres} \] - Container 2 (Mixture Composition): - Liquid A in \( y \) litres of mixture: \[ \frac{y}{4} \text{ litres} \] - Liquid B in \( y \) litres of mixture: \[ \frac{3y}{4} \text{ litres} \] Forming the Ratio: \[ \text{Liquid A : Liquid B} = \left[ \frac{5x}{6} + \frac{y}{4} \right] : \left[ \frac{x}{6} + \frac{3y}{4} \right] \] Given: \[ \frac{5x}{6} + \frac{y}{4} = \frac{x}{6} + \frac{3y}{4} \] Solving for \( x:y \): \[ \frac{5x}{6} - \frac{x}{6} = \frac{3y}{4} - \frac{y}{4} \] \[ \frac{4x}{6} = \frac{2y}{4} \] \[ \frac{2x}{3} = \frac{y}{2} \] \[ \frac{x}{y} = \frac{3}{4} \] Thus, the required ratio is: \[ x : y = 3 : 4 \]
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