# Mixtures

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## Mixtures: Basic Concept, Practice Questions

### FORMULAS:

If A and B are in the ratio m:n in a mixture so, whenever some quantity of the mixture is withdrawn, the withdrawn quantity will have A and B in the ratio m:n. If x litres is withdrawn then, in these x litres A = mx/(m+n) B = nx/(m+n)

### Repeated Dilution of a Mixture:

If a container initially contains V unit of liquid and x unit of liquid is taken out and it is filled with x unit of another liquid, then after n operations, the final quantity of the original liquid in the container is given as V(1-x/V)n

Q1) One man adds 6 litres of water to 11 litres of milk and 9 litres of water to 8 litres of milk. What is the ratio of the strengths of milk in the two mixtures?
1) 2:3
2) 3:2
3) 11:8
4) 8:11

Solution:

Required ratio = 11:8
Hence, option 3.

Q2) Two vessels contain equal quantity of mixtures of milk and water in the ratio 8:9 and 12:5 respectively. Both the mixtures are now mixed thoroughly. Find the ratio of milk to water in the new mixture so obtained.
1) 7:10
2) 10:7
3) 13:21
4) 21:13

Solution:

Vessel 1: 8:9 ; Vessel 2: 12:5
Volume of vessels = LCM of (8 + 9, 12 + 5) = 17 litres
For Vessel 1: Milk = 8 litres, Water = 9 litres
For Vessel 2: Milk = 12 litres, Water = 5 litres
On mixing, Milk = 8 + 12 = 20 litres, Water = 9 + 5 = 14 litres
Ratio of milk to water = 20:14 = 10:7
Hence, option 2.

Q3) The contents of two vessels containing water and milk are in the ratio 3:4 and 5:4 are mixed in the ratio 1:4. The resulting mixture will have water and milk in the ratio.
1) 184:176
2) 167:184
3) 167:148
4) 148:167

Solution:

Vessel 1: 3:4 ; Vessel 2: 5:4
LCM of (3 + 4, 5 + 4) = 63
As contents of two vessels are mixed in the ratio 1:4
For vessel 1: Volume = 63 litres Milk = (3/7)×63 = 27 litres, Water = 36 litres
For vessel 2: Volume = 252 litres Milk = (5/9)×252 = 140 litres, Water = 112 litres
On mixing, Milk = 27 + 140 = 167 litres, Water = 36 + 112 = 148 litres
Ratio of water to milk = 167:148
Hence, option 3.

Q4) In a mixture of 60 litres, the ratio of milk and water is 2:1. If the ratio of milk and water is to be 1:2, then the amount of water to be further addded is
1) 42 litres
2) 56 litres
3) 60 litres
4) 77 litres

Solution:

Milk = (2/3)×60 = 40 litres, Water = 20 litres
Let x litres of water be added to the mixture so, 40/(20 + x)= 1/2
On solving, x = 60 litres
Hence, option 3.

Q5) A mixture contains milk and water in the ratio of 9:4. On adding 4 litres of water, the ratio of milk to water becomes 3:2. Find the quantity of the original mixture.
1) 26 litres
2) 18 litres
3) 10 litres
4) 30 litres

Solution:

Milk = 9x, Water = 4x
9x/(4x + 4) = 3/2 i.e. x = 2
Quantity of the original mixture = 9x + 4x = 13x = 26 litres.
Hence, option 1.

Q6) A mixture contains milk and water in the ratio of 9:4. On adding 8 litres of water, the ratio of milk to water becomes 3:2. Find the quantity of the original mixture.
1) 26 litres
2) 52 litres
3) 104 litres
4) 30 litres

Solution:

Milk = 9x, Water = 4x
9x/(4x + 8) = 3/2 i.e. x = 4
Quantity of the original mixture = 9x + 4x = 13x = 52 litres.
Hence, option 2.

Q7) A bucket contains a mixture of two liquids A and B in the proportion 5:3. If 16 litres of the mixture is replaced by 16 litres of liquid B, then the ratio of the two liquids becomes 3:5. How much of the liquid B was there in the bucket.
1) 25 litres
2) 15 litres
3) 24 litres
4) 18 litres

Solution:

Initial mixture: A = 5x, B = 3x
In 16 litres of the mixture that is replaced
A = (5/8)×16 = 10 litres, B = 6 litres
Final mixture: A = 5x - 10, B = 3x - 6 + 16 = 3x + 10
(5x - 10)/(3x + 10) = 3/5 i.e. x = 5
Liquid B = 3(5) = 15 litres
Hence, option 4.

Q8) A vessel contains liquid A and B in the ratio 7:6. If 26 litres of the mixture are removed and the same quantity of liquid B is added, the ratio becomes 6:7. What quantity does the vessel hold?
1) 142 litres
2) 156 litres
3) 172 litres
4) 182 litres

Solution:

Initial mixture: A = 7x, B = 6x
In 26 litres of the mixture that is replaced
A = (7/13)×26 = 14 litres, B = 12 litres
Final mixture: A = 7x - 14, B = 6x - 12 + 26 = 6x + 14
(7x - 14)/(6x + 14) = 6/7 i.e. x = 14
Quantity the vessel holds = 13x = 13(14) = 182 litres
Hence, option 3.

Q9) From a cask of wine, containing 64 litres, 8 litres are drawn out and the cask is filled with water. If the same process is repeated a second, then a third time, what will be the proportion of wine to water in the resulting mixture?
1) 343:169
2) 343:512
3) 169:343
4) 512:343

Solution:

Volume of wine left = Initial volume of wine (1 - x/V)n
Initial volume of wine = 64 litres = V (Volume of the vessel)
x = 8 litres; n = 3 times
Volume of wine left = 64(1 - 8/64)3 = 343/8 litres
Ratio of wine to water in the resulting mixture = (343/8)/[64-(343/8)] = 343/169