Arithmetic

Mixture & Alligation

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 $2: 3$ ?

Entered answer:

✅ Correct Answer: 8

We start with 40 litres where dye and water are in the ratio 2:3.

When we have a ratio like 2:3, it means:

Out of every 5 parts of solution, 2 parts are dye and 3 parts are water

Total parts = 2 + 3 = 5 parts

Calculating actual quantities:

Dye = $\tfrac{2}{5} \times 40 = 16$ litres

Water = $\tfrac{3}{5} \times 40 = 24$ litres

Check: 16 + 24 = 40 litres

When we add water, the amount of dye stays the same (16 litres).

The new ratio is 2:5 (dye:water), which means:

Total parts = 2 + 5 = 7 parts

Dye = 2 parts, Water = 5 parts

Since dye remains 16 litres and represents 2 parts:

1 part = $\tfrac{16}{2} = 8$ litres

Total solution = 7 parts = $7 \times 8 = 56$ litres

Water = 5 parts = $5 \times 8 = 40$ litres

Water added = 40 - 24 = 16 litres

Amount removed = $\tfrac{1}{4} \times 56 = 14$ litres

Remaining solution = 56 - 14 = 42 litres

When we remove part of a mixture, we remove it in the same ratio as the original mixture.

Since the ratio is still 2:5:

Dye remaining = $\tfrac{2}{7} \times 42 = 12$ litres

Water remaining = $\tfrac{5}{7} \times 42 = 30$ litres

Current state: 12 litres dye, 30 litres water

Target: Ratio of 2:3 (dye:water)

We're only adding dye, so water remains 30 litres.

If the final ratio is 2:3 and water = 30 litres:

Water represents 3 parts = 30 litres

1 part = $\tfrac{30}{3} = 10$ litres

Dye should be 2 parts = $2 \times 10 = 20$ litres

Dye to be added = 20 - 12 = 8 litres

8 litres of dye must be added.

Always track what stays constant in each step:

Adding water: Dye stays constant while water is added

Removing solution: Ratio stays constant while solution is removed proportionally

Adding dye: Water stays constant while dye is added

This systematic approach prevents confusion and ensures accuracy.

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