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Mixtures & Alligation aptitude questions

Blending prices and concentrations; replacing part of a solution.

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Formula sheetMixtures & Alligation: the essentialsFormula sheetMixtures & Alligation: the essentialsShow

  • Alligation: quantity of cheaper : quantity of dearer = (dearer − mean) : (mean − cheaper). It works for prices, concentrations, wages and marks — any weighted average of two groups.
  • The mean sits nearer the ingredient there is more of; exactly midway means 1 : 1.
  • Weighted mean = (q₁p₁ + q₂p₂ + …) ÷ (q₁ + q₂ + …), for any number of ingredients.
  • Repeated replacement: from V litres, drawing off x litres and replacing it with water n times leaves V × (1 − x/V)ⁿ of the original liquid.
  • Dilution: the dissolved substance stays fixed. New total = that amount ÷ the new concentration; water to add = new total − old total.
  • Strengthening with the pure component: anchor on the part that is not being added, usually the water.
  • A milkman who adds water and sells at cost price gains water ÷ milk × 100 per cent.
  • For a profit target, turn the selling price into the cost the mixture must have, then apply alligation to that cost.
  • Mixing two mixtures: convert each to actual quantities first. For equal volumes, use the LCM of the ratio sums.
  • Anything drawn off a mixture removes both components in proportion.

Example: tea at ₹120 and ₹180 per kg mixed to ₹160: (180 − 160) : (160 − 120) = 20 : 40 = 1 : 2.

Questions37

1

Mixing rice of two prices

Rice at ₹40/kg is mixed with rice at ₹60/kg so that the mixture costs ₹52/kg. In what ratio are they mixed?

alligation

45s

Easy2

Adding water to a milk mixture

A 60-litre mixture contains milk and water in the ratio 2 : 1. How much water must be added to make the ratio 1 : 1?

milk and water

45s

Easy3

Replacing milk with water, twice

A vessel holds 100 L of milk. 10 L is drawn off and replaced with water, and this is done twice in all. How much milk remains?

replacement

45s

Medium4

Blending two alloys

Alloy A is 40% copper and alloy B is 70% copper. In what ratio should they be melted together to give an alloy that is 50% copper?

alloys

45s

Easy5

Skilled and unskilled workers

The average wage of 30 workers is ₹500. Skilled workers earn ₹600 and unskilled ₹450. How many workers are skilled?

weighted average

60s

Medium6

Diluting an acid solution

How much water must be added to 40 L of a 30% acid solution to make it a 20% solution?

dilution

45s

Easy7

Blending two varieties of rice

Rice at ₹30 per kg is mixed with rice at ₹45 per kg so that the mixture is worth ₹35 per kg. In what ratio are they mixed?

alligation

45s

Easy8

Blending tea for a target price

In what ratio must tea at ₹62 per kg be mixed with tea at ₹72 per kg so the mixture is worth ₹64.50 per kg?

alligation

60s

Medium9

Adding water to milk for profit

In what ratio must water be mixed with milk costing ₹32 per litre so that the seller gains 25% by selling the mixture at ₹32 per litre?

alligation · milk and water

120s

Hard10

Two 4-litre replacements in 40 litres of milk

A vessel holds 40 litres of milk. 4 litres is removed and replaced with water, and this is done twice in all. How much milk remains?

replacement

75s

Medium11

Three replacements

A cask holds 80 litres of wine. 8 litres is drawn off and replaced with water three times. How much wine is left?

replacement

75s

Medium12

Blending a 60% and a 20% copper alloy

Alloy A contains 60% copper and alloy B contains 20% copper. In what ratio must they be mixed to obtain an alloy with 45% copper?

alloys

60s

Medium13

Price of the mixture

15 kg of sugar at ₹40 per kg is mixed with 25 kg at ₹32 per kg. What is the price per kg of the mixture?

weighted average

60s

Easy14

The ratio after adding water

A 45-litre mixture has milk and water in the ratio 4 : 1. If 10 litres of water is added, what is the new ratio?

milk and water

75s

Medium15

Two vessels poured together

Two vessels of equal capacity hold milk and water in the ratios 5 : 3 and 5 : 7. Their contents are poured into a third vessel. What is the new ratio?

alloys

105s

Hard16

Water needed to reach a ratio

A 60-litre mixture contains milk and water in the ratio 7 : 3. How much water must be added so the ratio becomes 3 : 2?

milk and water

90s

Medium17

Strengthening an acid solution

How much pure acid must be added to 50 litres of a 20% acid solution to make it 50% acid?

concentration

120s

Hard18

Milkman's profit from water

A milkman mixes water with milk in the ratio 1 : 4 and sells the mixture at cost price. What is his profit percent?

milk and water

90s

Medium19

Blending two grades of wheat

Wheat at ₹24 per kg is mixed with wheat at ₹36 per kg so the mixture is worth ₹30 per kg. In what ratio are they mixed?

alligation

45s

Easy20

Water added for a 20% gain

A milkman mixes water with milk and sells the mixture at cost price, gaining 20%. What is the ratio of water to milk?

milk and water

75s

Medium21

One replacement

From 50 litres of milk, 10 litres is removed and replaced with water. How much milk remains?

replacement

45s

Easy22

Copper in two alloys

Alloy P is 30% copper and alloy Q is 60% copper. In what ratio must they be mixed to obtain 40% copper?

alloys

60s

Medium23

Strengthening a spirit solution

How much pure spirit must be added to 60 litres of a 40% spirit solution to make it 50% spirit?

concentration

105s

Hard24

Three quantities blended

10 kg at ₹20, 20 kg at ₹30 and 20 kg at ₹40 are mixed. What is the average price per kg?

weighted average

75s

Medium25

Draw off and replace

A vessel holds 20 litres of a mixture with milk and water in the ratio 3 : 1. 4 litres is drawn off and replaced with water. What is the new ratio?

replacement

120s

Hard26

Unequal quantities poured together

10 litres of a 3 : 2 milk-and-water mixture is poured together with 15 litres of a 2 : 3 mixture. What is the new ratio?

alloys

105s

Hard27

Two grades of worker

A firm pays skilled workers ₹800 a day and unskilled workers ₹500 a day. If the average daily wage is ₹600, what is the ratio of skilled to unskilled workers?

weighted average

75s

Medium28

Diluting a sugar solution

40 litres of a solution is 25% sugar. How much water must be added to reduce it to 20% sugar?

concentration

75s

Medium29

Mixing for a target selling price

In what ratio must tea at ₹80 per kg be mixed with tea at ₹120 per kg so that selling the mixture at ₹110 per kg gives a 10% profit?

alligation

105s

Hard30

Blending two acid solutions

A 20% acid solution is mixed with a 50% acid solution to give 30% acid. In what ratio are they mixed?

alligation

60s

Medium31

Adulterated fuel

A dealer mixes kerosene costing ₹40 per litre with petrol costing ₹100 per litre and sells the mixture at ₹100 per litre, gaining 25%.

alligation

120s

Hard32

Two replacements from a ratio

A cask holds 64 litres of milk. 8 litres is removed and replaced with water, twice. How much milk remains?

replacement

75s

Medium33

Rice at two prices

Rice costing ₹50 per kg is mixed with rice costing ₹70 per kg to give a mixture worth ₹56 per kg. In what ratio?

alligation

60s

Medium34

Water in a mixture

A 50-litre mixture contains 30% water. How much milk must be added to reduce the water to 20%?

concentration

90s

Hard35

Average cost of three items

5 kg at ₹40, 10 kg at ₹50 and 5 kg at ₹80 are mixed. What is the cost per kg of the mixture?

weighted average

75s

Medium36

Two groups of candidates

In an exam the average mark of passed candidates is 70 and of failed candidates 30. The overall average is 60. What is the ratio of passed to failed candidates?

weighted average

75s

Medium37

Repeated replacement

A tank holds 100 litres of milk. 20 litres is drawn off and replaced with water twice. How much milk remains?

replacement

60s

Medium

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Mixtures & Alligation Aptitude Questions with Solutions

37 Mixtures & Alligation questions with hints, worked solutions and time targets — 8 easy · 20 medium · 9 hard.

  • Section: Quantitative Aptitude
  • Questions: 37
  • By difficulty: 8 easy · 20 medium · 9 hard
  • Cost: Free, unlimited on every plan

Blending prices and concentrations; replacing part of a solution.

Mixtures & Alligation: the essentials

  • Alligation: quantity of cheaper : quantity of dearer = (dearer − mean) : (mean − cheaper). It works for prices, concentrations, wages and marks — any weighted average of two groups.
  • The mean sits nearer the ingredient there is more of; exactly midway means 1 : 1.
  • Weighted mean = (q₁p₁ + q₂p₂ + …) ÷ (q₁ + q₂ + …), for any number of ingredients.
  • Repeated replacement: from V litres, drawing off x litres and replacing it with water n times leaves V × (1 − x/V)ⁿ of the original liquid.
  • Dilution: the dissolved substance stays fixed. New total = that amount ÷ the new concentration; water to add = new total − old total.
  • Strengthening with the pure component: anchor on the part that is not being added, usually the water.
  • A milkman who adds water and sells at cost price gains water ÷ milk × 100 per cent.
  • For a profit target, turn the selling price into the cost the mixture must have, then apply alligation to that cost.
  • Mixing two mixtures: convert each to actual quantities first. For equal volumes, use the LCM of the ratio sums.
  • Anything drawn off a mixture removes both components in proportion.

Example: tea at ₹120 and ₹180 per kg mixed to ₹160: (180 − 160) : (160 − 120) = 20 : 40 = 1 : 2.

Questions

  1. Mixing rice of two prices Easy
    Rice at ₹40/kg is mixed with rice at ₹60/kg so that the mixture costs ₹52/kg. In what ratio are they mixed?
  2. Adding water to a milk mixture Easy
    A 60-litre mixture contains milk and water in the ratio 2 : 1. How much water must be added to make the ratio 1 : 1?
  3. Replacing milk with water, twice Medium
    A vessel holds 100 L of milk. 10 L is drawn off and replaced with water, and this is done twice in all. How much milk remains?
  4. Blending two alloys Easy
    Alloy A is 40% copper and alloy B is 70% copper. In what ratio should they be melted together to give an alloy that is 50% copper?
  5. Skilled and unskilled workers Medium
    The average wage of 30 workers is ₹500. Skilled workers earn ₹600 and unskilled ₹450. How many workers are skilled?
  6. Diluting an acid solution Easy
    How much water must be added to 40 L of a 30% acid solution to make it a 20% solution?
  7. Blending two varieties of rice Easy
    Rice at ₹30 per kg is mixed with rice at ₹45 per kg so that the mixture is worth ₹35 per kg. In what ratio are they mixed?
  8. Blending tea for a target price Medium
    In what ratio must tea at ₹62 per kg be mixed with tea at ₹72 per kg so the mixture is worth ₹64.50 per kg?
  9. Adding water to milk for profit Hard
    In what ratio must water be mixed with milk costing ₹32 per litre so that the seller gains 25% by selling the mixture at ₹32 per litre?
  10. Two 4-litre replacements in 40 litres of milk Medium
    A vessel holds 40 litres of milk. 4 litres is removed and replaced with water, and this is done twice in all. How much milk remains?
  11. Three replacements Medium
    A cask holds 80 litres of wine. 8 litres is drawn off and replaced with water three times. How much wine is left?
  12. Blending a 60% and a 20% copper alloy Medium
    Alloy A contains 60% copper and alloy B contains 20% copper. In what ratio must they be mixed to obtain an alloy with 45% copper?
  13. Price of the mixture Easy
    15 kg of sugar at ₹40 per kg is mixed with 25 kg at ₹32 per kg. What is the price per kg of the mixture?
  14. The ratio after adding water Medium
    A 45-litre mixture has milk and water in the ratio 4 : 1. If 10 litres of water is added, what is the new ratio?
  15. Two vessels poured together Hard
    Two vessels of equal capacity hold milk and water in the ratios 5 : 3 and 5 : 7. Their contents are poured into a third vessel. What is the new ratio?
  16. Water needed to reach a ratio Medium
    A 60-litre mixture contains milk and water in the ratio 7 : 3. How much water must be added so the ratio becomes 3 : 2?
  17. Strengthening an acid solution Hard
    How much pure acid must be added to 50 litres of a 20% acid solution to make it 50% acid?
  18. Milkman's profit from water Medium
    A milkman mixes water with milk in the ratio 1 : 4 and sells the mixture at cost price. What is his profit percent?
  19. Blending two grades of wheat Easy
    Wheat at ₹24 per kg is mixed with wheat at ₹36 per kg so the mixture is worth ₹30 per kg. In what ratio are they mixed?
  20. Water added for a 20% gain Medium
    A milkman mixes water with milk and sells the mixture at cost price, gaining 20%. What is the ratio of water to milk?
  21. One replacement Easy
    From 50 litres of milk, 10 litres is removed and replaced with water. How much milk remains?
  22. Copper in two alloys Medium
    Alloy P is 30% copper and alloy Q is 60% copper. In what ratio must they be mixed to obtain 40% copper?
  23. Strengthening a spirit solution Hard
    How much pure spirit must be added to 60 litres of a 40% spirit solution to make it 50% spirit?
  24. Three quantities blended Medium
    10 kg at ₹20, 20 kg at ₹30 and 20 kg at ₹40 are mixed. What is the average price per kg?
  25. Draw off and replace Hard
    A vessel holds 20 litres of a mixture with milk and water in the ratio 3 : 1. 4 litres is drawn off and replaced with water. What is the new ratio?
  26. Unequal quantities poured together Hard
    10 litres of a 3 : 2 milk-and-water mixture is poured together with 15 litres of a 2 : 3 mixture. What is the new ratio?
  27. Two grades of worker Medium
    A firm pays skilled workers ₹800 a day and unskilled workers ₹500 a day. If the average daily wage is ₹600, what is the ratio of skilled to unskilled workers?
  28. Diluting a sugar solution Medium
    40 litres of a solution is 25% sugar. How much water must be added to reduce it to 20% sugar?
  29. Mixing for a target selling price Hard
    In what ratio must tea at ₹80 per kg be mixed with tea at ₹120 per kg so that selling the mixture at ₹110 per kg gives a 10% profit?
  30. Blending two acid solutions Medium
    A 20% acid solution is mixed with a 50% acid solution to give 30% acid. In what ratio are they mixed?
  31. Adulterated fuel Hard
    A dealer mixes kerosene costing ₹40 per litre with petrol costing ₹100 per litre and sells the mixture at ₹100 per litre, gaining 25%.
  32. Two replacements from a ratio Medium
    A cask holds 64 litres of milk. 8 litres is removed and replaced with water, twice. How much milk remains?
  33. Rice at two prices Medium
    Rice costing ₹50 per kg is mixed with rice costing ₹70 per kg to give a mixture worth ₹56 per kg. In what ratio?
  34. Water in a mixture Hard
    A 50-litre mixture contains 30% water. How much milk must be added to reduce the water to 20%?
  35. Average cost of three items Medium
    5 kg at ₹40, 10 kg at ₹50 and 5 kg at ₹80 are mixed. What is the cost per kg of the mixture?
  36. Two groups of candidates Medium
    In an exam the average mark of passed candidates is 70 and of failed candidates 30. The overall average is 60. What is the ratio of passed to failed candidates?
  37. Repeated replacement Medium
    A tank holds 100 litres of milk. 20 litres is drawn off and replaced with water twice. How much milk remains?

Placement tests that ask mixtures & alligation