A solution, of volume 40 litres, has dye and water in the proportion 2 : 3. Water is added to the solution to change this proportion to 2 : 5. If one-fourths of this diluted solution is taken out, how many litres of dye must be added to the remaining solution to bring the proportion back to 2 : 3?
Approach Solution - 1
Original quantity of dye and water in the original solution i.e., \(16\) \(litres\) \(\bigg(i.e. = 40×\frac{2}{5}\bigg)\) and \(24\) \(litres\) \((i.e. = 40-16)\)
Quantity of water added = \(16\) \(litres\) (As \(1\) part = \(8\) \(litres\)).
Quantity of dye and water removed = \(\frac{1}{4}×\frac{2}{7}(56)\) i.e., \(4\) litres and \(\frac{1}{4}×\frac{5}{7}×(56)\) i.e., \(10\)l litres.
Final quantity of dye and water is \(12\) litres and \(30\) litres.
\(∴\) Quantity of dye to be added to make the ratio of dye and water again \(2: 3\) i.e., \(8\) litres.
Approach Solution -2
Initially, there are 16 liters of dye and 24 liters of water. To achieve a ratio of dye to water of 2:5, the amount of water should be increased to 40 liters for every 16 liters of dye. Consequently, 16 liters of water are added.
Now, the quantities of dye and water are 16 liters and 40 liters respectively. After removing \(\frac{1}{4}\)th of the solution, the amounts become 12 liters of dye and 30 liters of water.
To establish a dye-to-water ratio of 2:3, for every 30 liters of water, we need 20 liters of dye. This means 8 liters of dye should be added. Therefore, the correct answer is 8.
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