Alligation or Mixture problems
- 1. A jar full of whisky contains 40% alcohol. A part of this whisky is replaced by another containing 19% alcohol and now the percentage of alcohol was found to be 26%. The quantity of whisky replaced is:
Options- A.
1 3 - B.
2 3 - C.
2 5 - D.
3 5
Discuss
Correct Answer:
| 2 |
| 3 |
Explanation:
| Strength of first jar Strength of 2nd jar | ||
| 40% | Mean Strength 26% |
19% |
| 7 | 14 | |
So, ratio of 1st and 2nd quantities = 7 : 14 = 1 : 2
| ∴ Required quantity replaced = | 2 |
| 3 |
- 2. A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?
Options- A.
1 3 - B.
1 4 - C.
1 5 - D.
1 7
Discuss
Correct Answer:
| 1 |
| 5 |
Explanation:
Let x litres of this liquid be replaced with water.
| Quantity of water in new mixture = | ❨ | 3 - | 3x | + x | ❩ | litres |
| 8 |
| Quantity of syrup in new mixture = | ❨ | 5 - | 5x | ❩ | litres |
| 8 |
| ∴ | ❨ | 3 - | 3x | + x | ❩ | = | ❨ | 5 - | 5x | ❩ |
| 8 | 8 |
⟹ 5x + 24 = 40 - 5x
⟹ 10x = 16
| ⟹ x = | 8 | . |
| 5 |
| So, part of the mixture replaced = | ❨ | 8 | x | 1 | ❩ | = | 1 | . |
| 5 | 8 | 5 |
- 3. In what ratio must water be mixed with milk to gain 16⅔% on selling the mixture at cost price?
Options- A. 1 : 6
- B. 6 : 1
- C. 2 : 3
- D. 4 : 3 Discuss
Correct Answer: 1 : 6
Explanation:
| S.P. of 1 litre of mixture = Re.1, Gain = | 50 | %. |
| 3 |
| ∴ C.P. of 1 litre of mixture = | ❨ | 100 x | 3 | x 1 | ❩ | = | 6 |
| 350 | 7 |
By the rule of alligation, we have:
| C.P. of 1 litre of water C.P. of 1 litre of milk | |||||
| 0 | Mean Price
|
Re. 1 | |||
|
|
||||
| ∴ Ratio of water and milk = | 1 | : | 6 | = 1 : 6. |
| 7 | 7 |
- 4. A can contains a mixture of two liquids A and B is the ratio 7 : 5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7 : 9. How many litres of liquid A was contained by the can initially?
Options- A. 10
- B. 20
- C. 21
- D. 25 Discuss
Correct Answer: 21
Explanation:
| Quantity of A in mixture left = | ❨ | 7x - | 7 | x 9 | ❩ | litres = | ❨ | 7x - | 21 | ❩ litres. |
| 12 | 4 |
| Quantity of B in mixture left = | ❨ | 5x - | 5 | x 9 | ❩ | litres = | ❨ | 5x - | 15 | ❩ litres. |
| 12 | 4 |
| ∴ |
|
= | 7 | |||||
|
9 |
| ⟹ | 28x - 21 | = | 7 |
| 20x + 21 | 9 |
⟹ 252x - 189 = 140x + 147
⟹ 112x = 336
⟹ x = 3.
So, the can contained 21 litres of A.
- 5. 8 litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of water is 16 : 65. How much wine did the cask hold originally?
Options- A. 18 litres
- B. 24 litres
- C. 32 litres
- D. 42 litres Discuss
Correct Answer: 24 litres
Explanation:
| Then, quantity of wine left in cask after 4 operations = | [ | x | ❨ | 1 - | 8 | ❩ | 4 | ] litres. |
| x |
| ∴ | ❨ | x(1 - (8/x))4 | ❩ | = | 16 |
| x | 81 |
| ⟹ | ❨ | 1 - | 8 | ❩ | 4 | = | ❨ | 2 | ❩ | 4 |
| x | 3 |
| ⟹ | ❨ | x - 8 | ❩ | = | 2 |
| x | 3 |
⟹ 3x - 24 = 2x
⟹ x = 24.
- 6. In a mixture of 45 liters, the ratio of milk and water is 4:1. How much water must be added to make the mixture ratio 3:2?
Options- A. 72 litres
- B. 24lLitres
- C. 15 litres
- D. 1.5 litres Discuss
Correct Answer: 15 litres
Explanation:
Let us assume the ratio factor is x.
According to question,
Quantity of milk = 4x liters and Quantity of Water = x liters.
According to question,
4x + x = 45
? 5x = 45
? x = 9
Quantity of milk = 4x = 9 x 4 = 36 liters
Quantity of water = x = 9 liters
Let y liters of water be added to make the ratio 3:2
Then,
36/(9 + y) = 3/2
? 72 = 27 + 3y
? y = 15 liters
If you add 15 liters of water, the ratio will become 3:2.
- 7. In a class of 30 students , the average weight of boys is 20 kg and the average weight of the girls is 25 kg . The fraction of boys out of the total students of the class is
Options- A. 4/5
- B. 5/6
- C. 3/4
- D. Data insufficient Discuss
Correct Answer: Data insufficient
Explanation:
Let us assume the number of boys = B and number of girls = G.
According to question,
B + G = 30
Lets us assume total weight of boys = W1 and total weight of girls = W2
average weight of boys = total weight of boys/number of boys
total weight of boys/number of boys = 20
W1/B = 20
W1 = 20B
average weight of girls = total weight of girls/number of girls
25 = W2/G
W2 = 25G
Data is not sufficient to solve the equation.
since we do not know either the average weight of the whole class or the ratio of no. of boys to girls.
- 8. In what ratio should milk and water be mixed so that after selling the mixture at the cost price a profit of 50/3 % is made?
Options- A. 1:2
- B. 6:1
- C. 2:3
- D. 2:5 Discuss
Correct Answer: 6:1
Explanation:
Method 1 to solve the equation.
Let us assume Cost Price (C.P) of 1 liter milk be Y rupees.
According to question,
Selling Price (S.P) of 1 liter of mixture Y rupees. (Selling price should be same as Cost price)
Profit = 50/3 %
Let us assume Cost price of 1 Liter of Mixture of Water and Milk = C
Cost price of 1 Liter of Mixture of Water and Milk = Selling price of Mixture - Profit
C = Y - C x 50/3%
? C = Y - C x 50/3x100
? C = Y - C x 1/3x2
? C = Y - C/6
? Y = C + C/6
? Y = (6C + C)/6
? Y = (6C + C)/6
? Y = 7C/6
? C = 6Y/7
Since in Y rupees we will get 1 liter milk.
Hence in 1 rupees we will get 1/Y liter milk.
Hence in 6Y/7 rupees we will get 1 x 6Y/7 x Y liter milk.
Hence in 6Y/7 rupees we will get 6/7 liter milk.
We need 1 liter mixture of milk and water for sold on the same price, we need to mix the water.
So water quantity in mixture = 1 - 6/7 = 1/7
Ratio of Milk and water in mixture = quantity of Milk in Mixture/quantity of Water in Mixture
Ratio of Milk and water in mixture = 6/7 / 1/7
Ratio of Milk and water in mixture = 6 x 7 / 7
Ratio of Milk and water in mixture = 6 / 1
Ratio of Milk and water in mixture = 6 : 1
Method 2 to solve the question.
Let the original amount of milk be 1 liter and the cost price is 1 rupees per liter.
Cost price of milk = 1 rupees.
Selling price of Mixture = 1 rupees.
When the milk is mix with x liters of water milk remaining is 1- x.
Let us assume the cost price of milk in mixture is Y and given that the Sold Price is 1.
we should use the Profit formula in algebra,
Y + Y x 50/3 % = 1
Y + Y x 50/3 x 100 = 1
Y + Y x /3 x 2 = 1
7Y/6 = 1
Y = 6/7
Y = 6/7 is the cost price of Mixture for 1 liter.
Since water is free so there is not cost of water in mixture. So price of milk is 6/7 rs in mixture.
Since as per question,
In 1 rupees we will get 1 liter of milk.
hence in 6/7 rupees we will get 1 x 6/7 = 6/7 liter of milk.
Now we have to make 1 liter of mixture with water and milk.
quantity of Water + Quantity of Milk = 1 liter
quantity of Water + 6/7 = 1
quantity of Water = 1 - 6/7 = (7 - 6)/7
quantity of Water = 1/7
Ratio of milk and Water = Quantity of Milk / quantity of Water
Ratio of milk and Water = (6/7) / (1/7)
Ratio of milk and Water = (6/7) x (7/1)
Ratio of milk and Water = 6 x 7/ 7 x 1
Ratio of milk and Water = 6 :1
- 9. Two solution of 90% and 97% purity and mixed resulting in 21 liters of mixture of 94% purity. How much is the quantity of the first solution in the resulting mixture?
Options- A. 15 litres
- B. 12 litres
- C. 9 litres
- D. 6 litres Discuss
Correct Answer: 9 litres
Explanation:
Method 1 to solve the equation.
Let us assume the number of liters of the 90% purity solution = A
and the number of liters of the 97% purity solution = B.
According to question,
Since there are 21 liters of the solution,
A + B = 21 ...................... (1)
Since after mixing the two solutions the new mixture has 94% purity,
Concentrate of A + Concentrate of B = Concentrate of (A + B)
A x 90% + B x 94% = (A+ B) x 97%
? A x 90/100 + B x 97/100 = (A + B) x 94/100
? 90A + 97B = (A + B) x 94
? 90A + 97B = 94A + 94B
? 94A + 94B - 90A- 97B = 0
? 4A - 3B = 0 ........................(2)
Multiply the 3 with Equation (1) and add with Equation (2),
3A + 3B + 4A - 3B = 63 + 0
? 7A = 63
? A = 63/7 = 9
Put the value of A in Equation (1) , we will get
9 + B = 21
B = 21 - 9
B = 12
The first solution would be A = 9 liters.
Method 2 to solve the equation.
Hit and trail method.
94% is closer to 97% but barely meaning the mixtures will not be equal parts but will be slightly more of the higher purity. Quickly eliminate A and B. Out of the others 9 is the easy choice. If the other choices were closer to half this wouldn't work.
- 10. In what proportion must tea at Rs. 62 per kg be mixed with tea at Rs. 72 per kg in order to obtain the mixture worth Rs. 65 per kg?
Options- A. 4: 6
- B. 7: 3
- C. 2: 3
- D. 4: 7 Discuss
Correct Answer: 7: 3
Explanation:
Using Alligation rule,
(Quantity of cheaper tea) / (Quantity of dearer tea) = (d - m) / (m - c) = 7/3
Therefore, they must be mixed in the ratio of 7 : 3.
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