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Alligation or Mixture problems

1. Tea worth of Rs. 135/kg and Rs. 126/kg are mixed with a third variety in the ratio 1: 1 : 2. If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be____?
Options
A. Rs. 169.50
B. Rs.1700
C. Rs. 175.50
D. Rs. 180

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Correct Answer: Rs. 175.50

Explanation:

To solve the problem, let's define the prices and quantities of the different varieties of tea being mixed.

Given:

Price of the first variety of tea = Rs. 135/kg
Price of the second variety of tea = Rs. 126/kg
Ratio of the first, second, and third varieties = 1 : 1 : 2

Let the price of the third variety of tea be $x$ Rs./kg.

The total mixture is worth Rs. 153/kg. Since the ratio of the tea varieties is 1:1:2, we can assume the quantities of each type of tea in the mixture to be:

Quantity of the first variety = 1 unit
Quantity of the second variety = 1 unit
Quantity of the third variety = 2 units

The total cost of the mixture can be expressed as the weighted average cost of the individual varieties based on their respective quantities.

Let's compute the cost per unit for the mixture:

\text{Total cost} = (1 \text{ unit} \times 135 \text{ Rs./kg}) + (1 \text{ unit} \times 126 \text{ Rs./kg}) + (2 \text{ units} \times x \text{ Rs./kg})

The total quantity of the mixture is:

1 \text{ unit} + 1 \text{ unit} + 2 \text{ units} = 4 \text{ units}

The average price per kg of the mixture is given as Rs. 153. Therefore, we set up the equation:

\frac{135 + 126 + 2x}{4} = 153

First, simplify the numerator:

135 + 126 + 2x = 261 + 2x

Next, substitute this back into the equation and solve for $x$ :

\frac{261 + 2x}{4} = 153

Multiply both sides by 4 to clear the denominator:

261 + 2x = 612

Now, isolate $x$ :

2x = 612 - 261

2x = 351

x = \frac{351}{2}

x = 175.5

Thus, the price of the third variety of tea is Rs. 175.50 per kg.

2. The diluted wine contains only 8 liters of wine and the rest is water. A new mixture whose concentration is 30%, is to be formed by replacing wine. How many liters of mixture shall be replaced with pure wine if there was initially 32 liters of water in the mixture?
Options
A. 4
B. 5
C. 8
D. None of these

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3. The ratio of water and alcohol in two different containers is 2:3 and 4:5. In what ratio we are required to mix the mixtures of two containers in order to get the new mixture in which the ratio of alcohol and water be 7:5?
Options
A. 7:3
B. 5:3
C. 8:5
D. 2:7

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Correct Answer: 5:3

Explanation:

$W 1 : A 1 W 2 : A 2 . . . . . W N : A N$

2 : 3 4 : 5 5 : 7

$\frac{W 1}{W 1 + A 1} = \frac{2}{5} \frac{W 2}{W 2 + A 2} = \frac{4}{9} \frac{W N}{W N + A N} = \frac{5}{12}$

= 72/180

= 80/180

= 75/180

=> 5 : 3

Therefore, the ratio is 5: 3

4. In a pot, there is a mixture of milk and water in the ratio 4 : 5. If it is filled with an additional 8 litres of milk, the pot would be full and ratio of milk and water would become 6 : 5. Find the capacity of the pot?
Options
A. 11 lit
B. 22 lit
C. 33 lit
D. 44 lit

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5. Ratio of water and milk in a container is 2 : 3. If 40 liter mixture removed from the container and same quantity of milk is added to it then the ratio of water to milk becomes 1 : 4. Find the initial quantity of mixture?
Options
A. 75 lit
B. 80 lit
C. 85 lit
D. 90 lit

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6. From a container of wine, a thief has stolen 15 litres of wine and replaced it with same quantity of water. He again repeated the same process. Thus, in three attempts the ratio of wine and water became 343 : 169. The initial amount of wine in the container was:
Options
A. 75 litres
B. 100 litres
C. 150 litres
D. 120 litres

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Correct Answer: 120 litres

Explanation:

$\frac{w i n e (l e f t)}{w i n e (a d d e d)} = \frac{343}{169}$

It means $\frac{w i n e (l e f t)}{w i n e (i n i t i a l a m o u n t)} = \frac{343}{512}$ (since 343 + 169 = 512)

Thus, $343 x = 512 x {(1 - \frac{15}{k})}^{3}$

$\frac{343}{512} = {(\frac{7}{8})}^{3} = {(1 - \frac{15}{k})}^{3}$

$(1 - \frac{15}{k}) = \frac{7}{8} = (1 - \frac{1}{8})$

Thus the initial amount of wine was 120 liters.

7. A tin a mixture of two liquids A and B in the proportion 4 : 1. If 45 litres of the mixture is replaced by 45 litres of liquid B, then the ratio of the two liquids becomes 2 : 5. How much of the liquid B was there in the tin? What quantity does the tin hold?
Options
A. 58 l
B. 65 l
C. 50 l
D. 62 l

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8. In a mixture of milk and water the proportion of water by weight was 75%. If in 60 gm of mixture 15 gm water was added, what would be the percentage of water? (Weight in gm)
Options
A. 80%
B. 70%
C. 75%
D. 62%

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9. In a 48 ltr mixture, the ratio of milk and water is 5:3. How much water should be added in the mixture so as the ratio will become 3:5?
Options
A. 24 lit
B. 16 lit
C. 32 lit
D. 8 lit

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10. A jar was full with honey. A person used to draw out 20% of the honey from the jar and replaced it with sugar solution. He has repeated the same process 4 times and thus there was only 512 gm of honey left in the jar, the rest part of the jar was filled with the sugar solution. The initial amount of honey in the jar was filled with the sugar solution. The initial amount of honey in the jar was:
Options
A. 1.25 kg
B. 1 kg
C. 1.5 kg
D. None of these

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