Mixtures & Alligations • Topic 3 of 4

Concentration & Dilution

Dilution problems track one fact: the amount of PURE substance does not change when you add water (the diluent), but the total volume grows — so the concentration falls. Lock onto the pure quantity. If a vessel holds V litres at c% concentration, the pure part is V×c/100; after adding w litres of water the new concentration is pure/(V + w) × 100. Strengthening works the same way in reverse — adding pure substance keeps the water fixed while raising the percentage. Two CAT-favourite phrasings to watch: "the concentration is reduced to one-third" (set up new pure-fraction = old/3 and solve for water added) and "make a 40% solution from a 60% one" (alligation between 60% and 0% water works beautifully here). Always decide first which component is conserved — pure or water — and base every equation on that fixed quantity rather than on the changing total.

✅ Solved examples

1. 20 litres of a 25% alcohol solution is diluted by adding 5 litres of water. Find the new concentration.
Alcohol = 20×0.25 = 5 L, unchanged. New total = 25 L ⇒ 5/25 = 20% alcohol.
2. How much water must be added to 30 litres of a 40% acid solution to make it 30%?
Acid = 12 L (fixed). Need 12/(30 + w) = 0.30 ⇒ 30 + w = 40 ⇒ w = 10 litres.
3. A 50-litre mixture is 90% milk. How much milk must be removed and replaced by water to make it 75% milk?
Use alligation between 90% and 0% (water) for the 75% target on 50 L: water fraction = (90−75)/90 = 1/6 ⇒ replace 50×1/6 = 8⅓ litres.
4. A vessel has 18 litres of 100% pure juice. How much water to add to bring it to 60% concentration?
Juice = 18 L fixed. 18/(18 + w) = 0.60 ⇒ 18 + w = 30 ⇒ w = 12 litres.

✏️ Practice — try these, take hints as needed

1. 40 L of 30% acid is diluted with 10 L water. New concentration?
Acid amount is unchanged.
Acid = 12 L, new total 50 L.
12/50 × 100.
24%
2. How much water to add to 24 L of 50% solution to make it 40%?
Pure = 12 L fixed.
12/(24 + w) = 0.40.
24 + w = 30.
6 litres
3. 60 L of milk is 80% pure. Water added to make it 60% pure?
Milk = 48 L fixed.
48/(60 + w) = 0.60.
60 + w = 80.
20 litres
4. A 90% sugar syrup (45 L). How much pure sugar to add to make it 92%? (sugar is the pure part)
Water = 4.5 L stays fixed.
After adding s: water fraction = 8%.
4.5/(45 + s) = 0.08.
11.25 litres
5. A 25 L mixture is 60% milk. Replace some with water to make it 48% milk. Amount replaced?
Alligation 60% vs 0% for target 48%.
water fraction = (60−48)/60 = 1/5.
25 × 1/5.
5 litres

📝 Topic test — 8 questions

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