Question:medium

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? [This Question was asked as TITA]

Updated On: Jan 15, 2026
  • 8 litres
  • 5 litres
  • 6 litres
  • 7 litres
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The Correct Option is A

Solution and Explanation

Initial Ratio of Dye to Water: $2:3$

Represent dye as $2x$ and water as $3x$.

Total Solution: $2x + 3x = 5x = 40$ litres

Solving for x: $x = 8$

  • Dye Amount: $2x = 2 \times 8 = 16$ litres
  • Water Amount: $3x = 3 \times 8 = 24$ litres

Water is added to achieve a new ratio of $2:5$, with the dye amount remaining at $16$ litres.

Let the new water quantity be $w$:

$\frac{16}{16 + w} = \frac{2}{5} \Rightarrow 5 \cdot 16 = 2(w + 16)$
$80 = 2w + 32 \Rightarrow 2w = 48 \Rightarrow w = 24$

Therefore, new water volume = 24 (added) + 24 (original) = 48 litres

New Total Volume = 16 (dye) + 48 (water) = 64 litres

One-fourth of the solution is removed: $\frac{1}{4} \cdot 64 = 16$ litres

The removed 16 litres, maintaining the $2:5$ ratio, consists of:

  • Dye Removed: $\frac{2}{7} \cdot 16 = 4.571$ litres (approximately 4.57 L)
  • Water Removed: $16 - 4.571 = 11.429$ litres (approximately 11.43 L)

Remaining solution:

  • Dye Remaining: $16 - 4.571 = 11.429$ L
  • Water Remaining: $48 - 11.429 = 36.571$ L

The objective is to achieve a new ratio of $2:3$:

Let the additional dye to be added be $y$, such that:

$\frac{11.429 + y}{36.571} = \frac{2}{3}$
$3(11.429 + y) = 2 \cdot 36.571$
$34.286 + 3y = 73.143 \Rightarrow 3y = 38.857 \Rightarrow y \approx 12.95$

Alternatively, using simplified ratio segments from earlier calculations:

  • After removing 1/4 of 64 L (which is 16 L), the remaining dye is $12$ L, and the remaining water is $30$ L.
  • To obtain a $2:3$ ratio with $30$ L of water, the required dye amount is $20$ L.
  • The extra dye to add is $20 - 12 = 8$ litres.

✅ Final Answer: 8 litres of dye must be added.

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