BANK & INSURANCE (MIXTURE AND ALLIGATION) PART 3

Total Questions: 30

1. Ratio of quantity of milk and water in a __ litre mixture (milk + water) is 5:3, respectively. 25% of this mixture is replaced with __ litres of water. This way the final quantity of water in the mixture becomes __ than that of milk in it. The values given in which of the following options will fill the blanks in the same order in which it is given to make the statement true:

I. 176, 16.5, 20% more
II. 112, 18.5, (100/21)% less
III. 160, 42, 16% more
IV. 220, 33, 8% less

Correct Answer: (d) Only (II), (III) and (IV)
Solution:

From (I) -
Initial quantity of milk in the mixture = {176 × (5/8)}
= 110 litres
And initial quantity of water in the mixture = (176 - 110) = 66 litres

Quantity of milk left in the mixture after taking out 25% of milk = 0.75 × 110 = 82.5 litres
Quantity of water left in the mixture after taking out 25% of water = 0.75 × 66 = 49.5 litres

Final quantity of water in the mixture = 49.5 + 16.5
= 66 litres

Required percentage = {(82.5 - 66)/82.5} × 100
= 20% less

So, I is false.

From (II) -
Initial quantity of milk in the mixture = {112 × (5/8)}
= 70 litres
And initial quantity of water in the mixture = (112 - 70) = 42 litres

Quantity of milk left in the mixture after taking out 25% of milk = 0.75 × 70 = 52.5 litres
Quantity of water left in the mixture after taking out 25% of water = 0.75 × 42 = 31.5 litres

Final quantity of water in the mixture = 31.5 + 18.5
= 50 litres

Required percentage = {(52.5 - 50)/52.5} × 100
= (100/21)% less

So, II is true.

From (III) -
Initial quantity of milk in the mixture = {160 × (5/8)}
= 100 litres
And initial quantity of water in the mixture = (160 - 100) = 60 litres

Quantity of milk left in the mixture after taking out 25% of milk = 0.75 × 100 = 75 litres
Quantity of water left in the mixture after taking out 25% of water = 0.75 × 60 = 45 litres

Final quantity of water in the mixture = (45 + 42)
= 87 litres

Required percentage = {(87 - 75)/75 × 100}
= 16% more

So, III is true.

From (IV) -
Initial quantity of milk in the mixture = {220 × (5/8)}
= 137.5 litres
And initial quantity of water in the mixture = (220 - 137.5) = 82.5 litres

Quantity of milk left in the mixture after taking out 25% of milk = 0.75 × 137.5 = 103.125 litres
Quantity of water left in the mixture after taking out 25% of water = 0.75 × 82.5 = 61.875 litres

Final quantity of water in the mixture = 61.875 + 33
= 94.875 litres

Required percentage = {(103.125 - 94.875)/103.125 × 100} = 8%

So, IV is true.

Hence, option d.

2. A shopkeeper has three mixtures (milk + water) i.e ‘A’ ‘B’ and ‘C’. The shopkeeper mixed 300 ml of mixture ‘A’ with 480 ml of mixture ‘B’ containing milk and water in the ratio of 5:7, respectively and formed mixture ‘P’. Now, 390 ml of mixture ‘P’ is taken out and mixed with 240 ml of mixture ‘C’ containing 50% more milk than water in it such that ratio of milk to water in the resultant mixture becomes 18:17, respectively. Find the ratio of milk to water in mixture A:

Correct Answer: (c) 8:7
Solution:Let quantity of milk in mixture ‘A’ be ‘x’ ml
So, quantity of water in mixture ‘A’ = (300 - x) ml

Quantity of milk in mixture ‘B’ = (5/12) × 480 = 200 ml
Quantity of water in mixture ‘B’ = 480 - 200 = 280 ml

Quantity of water in mixture ‘C’ = 240/2.5 = 96 ml
Quantity of milk in mixture ‘C’ = 240 - 96 = 144 ml

ATQ;
{[(200 + x)/780 × 390 + 144]}/{[(580 - x)/780 × 390 + 96]} = 18/17

{(200 + x)/2 + 144}/{(580 - x)/2 + 96} = 18/17

(x + 200 + 288)/(580 - x + 192) = 18/17

(x + 488)/(772 - x) = 18/17

17x + 8296 = 13896 - 18x

35x = 5600

x = 160

Desired ratio = 160 : (300 - 160) = 160 : 140 = 8 : 7

3. In 540 litres mixture of juice and water, the ratio of the quantity of juice to water is 4:1. 45 litres of the mixture is taken out and 36 litres of the water is added in the mixture. Again __ litres of the mixture is taken out and __ litres of the water added in the mixture, the ratio of the juice to water becomes 8:3.

The values given in which of the following options will fill the blanks in the same order in which it is given to make the statement true:
I. 45 litres, 24 litres
II. 50 litres, 20 litres
III. 59 litres, 12 litres
IV. 45 litres, 30 litres

Correct Answer: (b) Only III 
Solution:Initial quantity of juice in the mixture = 540 × 4/5
= 432 litres
Initial quantity of water in the mixture = 540 × 1/5
= 108 litres

After replacement,
Quantity of the juice in the mixture = 432 - 45 × 4/5
= 396 litres
Quantity of the water in the mixture = 108 - 45 × 1/5

  • 36 = 135 litres

So, the ratio of juice to water = 396:135 = 44:15

Again after replacement,

From I:
Ratio of the juice to water = (396 - 45 × 44/59) : (135 - 45 × 15/59 + 24)
= (396 - 1980/59) : (159 - 675/59)
= 21384/59 : 8706/59 = 10692:4353 = 3564:1451
It could not be the answer.

From II:
Ratio of the juice to water = (396 - 50 × 44/59) : (135 - 50 × 15/59 + 20)
= (396 - 2200/59) : (155 - 750/59)
= 21164/59 : 8395/59 = 21164:8395
It could not be the answer.

From III:
Ratio of the juice to water = (396 - 59 × 44/59) : (135 - 59 × 15/59 + 12)
= (396 - 44) : (147 - 15)
= 352:132 = 8:3
It could be the answer.

From IV:
Ratio of the juice to water = (396 - 45 × 44/59) : (135 - 45 × 15/59 + 30)
= (396 - 1980/59) : (165 - 675/59)
= 21384:9060 = 1782:755
It could not be the answer.

Hence, option b.

4. 480 ml of mixture ‘A’ contains __% more water than milk in it. If 60% of this mixture ‘A’ is taken out and is mixed with 360 ml of mixture ‘B’ containing 50% more water than milk in it. Further 540 ml of mixture is taken out of the resultant mixture and the remaining mixture is mixed with 384 ml of mixture ‘C’ containing milk and water in the ratio of 9:7, respectively, so that the difference between quantity of milk and water in the mixture finally is __ ml.

The values given in which of the following options will fill the blanks in the same order in which it is given to make the statement true:
I. 66(2/3)%, 24
II. 40%, 30
III. 18(2/11)%, 32

Correct Answer: (c) Only I and III
Solution:Quantity of milk in mixture ‘B’ = (2/5) × 360
= 144 ml
Quantity of water in mixture ‘B’ = 360 - 144
= 216 ml

Quantity of milk in mixture ‘C’ = (9/6) × 384
= 216 ml
Quantity of water in mixture ‘C’ = 384 - 216
= 168 ml

For ‘I’:
Quantity of milk in mixture ‘A’ = (3/8) × 480
= 180 ml
Quantity of water in mixture ‘A’ = 480 - 180 = 300 ml

When 60% of mixture ‘A’ is mixed with mixture ‘B’,
then ratio of milk to water in the resulting mixture =
(0.60 × 180 + 144) : (0.60 × 300 + 216)
= 252:396 = 7:11

Quantity of milk in 540 ml of mixture = (7/18) × 540
= 210 ml
Quantity of water in 540 ml of mixture
= 540 - 210 = 330 ml

Desired difference = (252 - 210 + 216) - (396 - 330 + 168) = 24 ml

So, ‘I’ can be true.

For ‘II’:
Quantity of milk in mixture ‘A’ = (5/12) × 480
= 200 ml
Quantity of water in mixture ‘A’ = 480 - 200 = 280 ml

When 60% of mixture ‘A’ is mixed with mixture ‘B’,
then ratio of milk to water in the resulting mixture =
(0.60 × 200 + 144) : (0.60 × 280 + 216)
= 264:384 = 11:16

Quantity of milk in 540 ml of mixture = (11/27) × 540 = 220 ml
Quantity of water in 540 ml of mixture
= 540 - 220 = 320 ml

Desired difference = (264 - 220 + 216) - (384 - 320 + 168) = 28 ml

So, ‘II’ cannot be the true.

For ‘III’:
Quantity of milk in mixture ‘A’ = (11/24) × 480
= 220 ml
Quantity of water in mixture ‘A’ = 480 - 220
= 260 ml

When 60% of mixture ‘A’ is mixed with mixture ‘B’,
then ratio of milk to water in the resulting mixture =
(0.60 × 220 + 144) : (0.60 × 260 + 216)
= 276:372 = 23:31

Quantity of milk in 540 ml of mixture
= (23/54) × 540 = 230 ml
Quantity of water in 540 ml of mixture
= 540 - 230 = 310 ml

Desired difference = (276 - 230 + 216) - (372 - 310 + 168) = 32 ml

So, ‘III’ can be true.
Hence, option c.

5. Container ‘P’ contains ‘X’ litres of mixture of milk and water in the ratio 8:5, respectively. When 25% of the mixture of container ‘P’ is replaced with ‘A’ litres of water, the ratio of milk to water becomes 6:5.

Container ‘Q’ contains (X – 56) litres of the mixture of milk to water in the ratio 5:3. After adding ‘(A – 6)’ litres of water to container ‘Q’, the ratio of milk to water becomes 4:3.
Which of the following statement/s can be determined from the given information?
I. The quantity of the mixture in container ‘Q’ initially.
II. The total quantity of water in mixtures of both containers ‘P’ and ‘Q’, initially.
III. The quantity of water that is added in container ‘P’.
IV. The quantity of milk that is taken out from container ‘P’.

Correct Answer: (d) All I, II, III and IV
Solution:The quantity of the mixture of container ‘P’ = ‘X’ litres
After taking out 25% of the mixture, the quantity of the mixture = ‘0.75X’ litres

According to the question,
[(0.75X/13) × 8] : [(0.75X/13) × 5 + A] = 6:5

5[(0.75X/13) × 8] = 6[(0.75X/13) × 5 + A]

30X/13 = (22.5X/13) + 6A

30X/13 - 22.5X/13 = 6A

7.5X = 78A

A = 7.5X/78 ……1

The quantity of the mixture of container ‘Q’
= (X - 56) litres

According to the question,
{[(X - 56)/8] × 5} : {[(X - 56)/8 × 3] + (A - 6)}
= 4:3

3{[(X - 56)/8] × 5} = 4{[(X - 56)/8 × 3] + (A - 6)}

(15X/8) - 105 = (12X/8) - 84 + 4A - 24

(15X/8) - (12X/8) = 105 - 84 - 24 + 4A

3X/8 = 4A - 3

3X/8 = 4(7.5X/78) - 3 (From equation 1)

3X/8 = 5X/13 - 3

5X/13 - 3X/8 = 3

X/104 = 3

X = 312

After putting the value of X in equation 1, we get
A = 7.5X/78
A = 7.5 × 312/78 = 30

For I:
The quantity of the mixture in container ‘Q’ = 312 – 56 = 256 litres
Therefore, I can be determined.

For II:
The total quantity of water in mixtures of both containers ‘P’ and ‘Q’ initially = 120 + 96 = 216 litres
Therefore, II can be determined.

For III:
The quantity of water that is added in container ‘P’ = 30 litres
Therefore, III can be determined.

For IV:
The quantity of milk that is taken out from container ‘P’ = 312 × 0.25 × (8/13) = 48 litres
Therefore, IV can be determined.

Hence, option d.

6. Directions (6-7): Answer the questions based on the information given below.

There are three mixtures ‘A’, ‘B’ and ‘C’ each of them contains milk and water. Ratio of quantity of milk to water in mixture ‘A’ is 5:2 such that quantity of milk in mixture ‘C’ is 20% more than that in mixture ‘A’. Ratio of quantity of milk and water in mixture ‘B’ is 3:5, respectively such that quantity of water in mixture ‘C’ is 20% less than that in mixture ‘B’. Difference between quantity of milk and water in mixture ‘C’ is 80 litres and quantity of milk in mixture ‘B’ is 50 litres more than that in mixture ‘A’.

Ques. Which of the following can be difference between quantity of milk in mixture ‘A’ and quantity of water in mixture ‘C’? (Note: m = 5 and p = 12)

Correct Answer: (c) 5p + 8m
Solution:Let quantity of milk and water in mixture ‘A’ be 5x litres and 2x litres respectively
Quantity of milk in mixture ‘C’ = 1.2 × 5x = 6x litres

Let quantity of milk and water in mixture ‘B’ be 3y litres and 5y litres respectively
Therefore, quantity of water in mixture ‘C’ = 0.8 × 5y = 4y litres

Case 1: In mixture ‘C’ let quantity of milk be more than that of water
Therefore, 6x – 4y = 80..... (1)
And, 3y – 5x = 50..... (2)
On solving equation (1) and (2), we get
Value of ‘x’ = -220 (not possible)

Therefore,
Case 2: In mixture ‘C’, quantity of milk is less than that of water
4y – 6x = 80..... (3)
3y – 5x = 50..... (4)
On solving equation (3) and (4), we get
Value of ‘x’ = 20 and value of y = 50

Quantity of milk in mixture ‘A’ = 5x = 100 litres
Quantity of water in mixture ‘A’ = 2x = 40 litres

Quantity of milk in mixture ‘B’ = 3y = 150 litres
Quantity of water in mixture ‘B’ = 5y = 250 litres

Quantity of milk in mixture ‘C’ = 6x = 120 litres
Quantity of water in mixture ‘C’ = 4y = 200 litres

Required difference = 200 – 100 = 100 litres = 5x
12 + 8 × 5 = 12p + 8m

Hence, option c.

7. Find the ratio of quantity of milk in given three mixtures.

Correct Answer: (c) 10:15:12
Solution:

Let quantity of milk and water in mixture ‘A’ be 5x litres and 2x litres respectively
Quantity of milk in mixture ‘C’ = 1.2 × 5x = 6x litres

Let quantity of milk and water in mixture ‘B’ be 3y litres and 5y litres respectively
Therefore, quantity of water in mixture ‘C’ = 0.8 × 5y = 4y litres

Case 1: In mixture ‘C’ let quantity of milk be more than that of water

Therefore, 6x – 4y = 80..... (1)
And, 3y – 5x = 50..... (2)
On solving equation (1) and (2), we get
Value of ‘x’ = -220 (not possible)

Therefore,
Case 2: In mixture ‘C’, quantity of milk is less than that of water
4y – 6x = 80..... (3)
3y – 5x = 50..... (4)
On solving equation (3) and (4), we get
Value of ‘x’ = 20 and value of y = 50

Quantity of milk in mixture ‘A’ = 5x = 100 litres
Quantity of water in mixture ‘A’ = 2x = 40 litres

Quantity of milk in mixture ‘B’ = 3y = 150 litres
Quantity of water in mixture ‘B’ = 5y = 250 litres

Quantity of milk in mixture ‘C’ = 6x = 120 litres
Quantity of water in mixture ‘C’ = 4y = 200 litres

Required ratio = 100:150:120 = 10:15:12

Hence, option c.

8. Mixture ‘A’ (milk and water) contains milk and water in the ratio 3:2, respectively, while mixture ‘B’ (milk and water) contains milk and water in the ratio of 4:3, respectively. When 25% of mixture ‘A’ is mixed with 75% of mixture ‘B’, in container ‘P’, the ratio of milk to water in the resultant mixture in container P becomes 18:13. Which of the following statements is true according to the question?

I. Quantity of milk in mixture ‘A’ is 40 litres more than that in ‘B’.
II. The sum of quantities of water in both the mixture is 70 litres.
III. Total quantity of mixture ‘A’ is more than that of ‘B’.

Correct Answer: (e) Only III
Solution:

Let the quantity of milk and water in mixture ‘A’ be 3x litres and 2x litres, respectively.
Let the quantity of milk and water in mixture ‘B’ be 4y litres and 3y litres, respectively.

According to the question,
(0.25 × 3x + 0.75 × 4y)/(0.25 × 2x + 0.75 × 3y)
= 18/13

9.75x + 39y = 9x + 40.5y

0.75x = 1.5y

x = 2y

For I:
Since, we cannot determine the quantity of milk in mixture ‘A’ and mixture ‘B’.
So, ‘I’ cannot be the answer.

For II:
Since, we cannot determine the quantity of water in both mixtures
So, ‘II’ cannot be the answer.

For III:
Total quantity of mixture ‘A’ = 5x = 5 × 2y = 10y litres
Total quantity of mixture ‘B’ = 4y + 3y = 7y litres

Since, 10y > 7y
So, ‘III’ can be the answer.

Hence, option e.

9. Directions (9-10): Answer the questions based on the information given below.

A vessel contains mixture of milk and water mixed in the ratio of 12:7 respectively. 76 liters of the mixture is taken out of the vessel and is replaced with ‘x’ liters of water, so that the ratio of the milk to water in the vessel becomes 8:x respectively. A milkman bought this final mixture in the vessel, but before buying he performed the purity test and paid only for the quantity of pure milk present in the mixture. He then sold the whole mixture at the rate 20% higher than the price of pure milk. In this transaction the milk man had the profit of 95%.

Ques. Find the value of ‘x’.

Correct Answer: (c) 5
Solution:

Let the final quantities of milk and water in the vessel be ‘8y’ litres and ‘xy’ litres respectively.

Let the cost price of 1 liter of pure milk = Rs. 10
Cost price of the mixture for the milkman = 8y × 10 = Rs. 80y

Selling price of 1 liter of mixture = 10 × 1.2 = Rs. 12
Selling price of the mixture of the milk = 80y × 1.95

= (8y + xy) × 12

156y = 96y + 12xy

12xy = 60y, x = 5

So, the value of x = 5

Hence, option c.

10. Find the difference in the initial quantities of milk and water in the vessel.

Correct Answer: (b) 70 liters 
Solution:

Let the final quantities of milk and water in the vessel be ‘8y’ litres and ‘xy’ litres respectively.

Let the cost price of 1 liter of pure milk = Rs. 10
Cost price of the mixture for the milkman = 8y × 10
= Rs. 80y

Selling price of 1 liter of mixture = 10 × 1.2 = Rs. 12
Selling price of the mixture of the milk = 80y × 1.95
= (8y + xy) × 12

156y = 96y + 12xy

12xy = 60y

x = 5

Let the initial quantities of milk and water in the vessel be ‘12z’ litres and ‘7z’ litres respectively.

76 liters of mixture contains 48 liters of milk and 28 liters of water.

So according to question,
(12z - 48)/(7z - 28 + 5) = 8/5

60z - 240 = 56z - 184

4z = 56, z = 14

So, the difference between the initial quantities of milk and water in the vessel = 5z = 14 × 5 = 70 liters

Hence, option b.