Alligation at the Juice Counter
A juice stall sells two concentrates. Concentrate X costs 80 taka per litre and is 60% pure fruit. Concentrate Y costs 50 taka per litre and is 30% pure fruit. The owner, Mizan, wants to prepare 60 litres of a mixed concentrate that is 45% pure fruit, and sell it at a price that earns 25% profit on his cost. He also has 10 litres of a leftover mix that is 40% pure fruit, which he wants to blend into the new batch without changing the final 45% purity. Separately, he wants to know how much water (0% fruit) can be added to 40 litres of the 45% mix so that the purity drops to 36%.
1. Using alligation, find how many litres of X and Y Mizan needs to make 60 litres of 45% pure mix.
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Alligation compares the purity percentages against the mean (45%).
- X is 60%: deviation above mean = 60 − 45 = 15.
- Y is 30%: deviation below mean = 45 − 30 = 15.
Ratio X : Y = (mean − Y) : (X − mean) = (45 − 30) : (60 − 45) = 15 : 15 = 1 : 1.
Total 60 litres in ratio 1:1 gives 30 litres of X and 30 litres of Y.
Check purity: (30 × 0.60 + 30 × 0.30)/60 = (18 + 9)/60 = 27/60 = 0.45 = 45%. Correct.
2. Find the cost of the 60 litres and the selling price per litre for 25% profit.
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Cost of X = 30 × 80 = 2400 taka.
Cost of Y = 30 × 50 = 1500 taka.
Total cost = 2400 + 1500 = 3900 taka.
Cost per litre = 3900/60 = 65 taka.
Selling price per litre for 25% profit = 65 × 1.25 = 81.25 taka.
Total revenue = 60 × 81.25 = 4875 taka; profit = 4875 − 3900 = 975 taka; 975/3900 = 25%. Correct.
3. Mizan blends his 10 litres of 40% leftover mix into the new batch. How much extra X or Y must he add so the final batch stays at 45% purity, and how much water can be added to 40 litres of 45% mix to bring it to 36%?
Read the worked answer
Part 1 - correcting the leftover blend. The 10 L of 40% mix contains 4 L pure fruit. Target is 45%. Let him add x litres of X (60%).
(4 + 0.60x)/(10 + x) = 0.45 4 + 0.60x = 4.5 + 0.45x 0.15x = 0.5 → x = 3.33 litres of X (approx).
Check: pure = 4 + 2 = 6; total = 13.33; 6/13.33 ≈ 0.45. Correct. So he adds about 3.33 L of X and no Y; the batch becomes 13.33 L at 45%.
Part 2 - diluting with water. 40 L of 45% mix has 18 L pure fruit. Adding water (0% fruit) keeps pure fruit at 18 L but raises total volume to 18/0.36 = 50 L.
Water to add = 50 − 40 = 10 litres.
Check: 18/50 = 0.36 = 36%. Correct.