Formulae
1. Alligation:
It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of desired price.
2. Mean Price:
The cost of a unit quantity of the mixture is called the mean price.
3. Rule of Alligation:
If two ingredients are mixed, then
\( {{Quantity of cheaper} \over {Quantity of dearer}} \) = \( {{C.P. of dearer - Mean Price} \over {Mean price - C.P. of cheaper}} \)
We present as under:
∴(Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c).
4. Suppose a container contains x of liquid from which y units are taken out and replaced by water.
After n operations, the quantity of pure liquid = \( [x {({1 - {y \over x}})^n}] \) units.
Exercise : Alligation or Mixture MCQ Questions and Answers
Question 1
An alloy contains zinc, copper and tin in the ratio 2 : 3 : 1 and another contains copper, tin and lead in the ratio 5 : 4 : 3. If equal weights of both alloys are melted together to form a third alloy, then the weight of lead per kg in new alloy will be:
| A. |
| B. |
| C. |
| D. |
Correct Answer : B. \( {1 \over 8} \) kg
Description :
Ratio of Zinc, Copper and Tin is given as,
Z : C : T = 2 : 3 : 1.
Now, let the first alloy be 12 kg (taken as 4 kg Zinc, 6 kg Copper and 2 Kg Lead). Weight of second alloy = 12 kg as, C : T : L = 5 : 4 : 3. (taken as 5 kg Copper, 4 kg Tin and 3 Kg Lead.)
Alloys are mixed together to form third alloy. Then the ratio of content in it,
Z : C : T : L = 4 : (6 + 5) : (2 + 4) : 3
Weight of third alloy = 12 + 12 = 24 Kg.
So, weight of the Lead = \( {3 \over 24} \) = \( {1 \over 8} \) kg.
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Question 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?
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| B. |
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Correct Answer : B. \( 1 \over 5 \)
Description :
Suppose the vessel initially contains 8 litres of liquid.
Let x litres of this liquid be replaced with water.
Quantity of water in new mixture = (3 - \( { 3x \over 8 } \) + x) litres
Quantity of syrup in new mixture = (5 - \( 5x \over 8 \)) litres
∴ ( 3 - \( 3x \over 8 \) +8 ) = (5 - \( 5x \over 8 \) )
⇒ 5x + 24 = 40 - 5x
⇒ 10x = 16
⇒ x = \( 8 \over 5 \)
So, part of the mixture replaced = \( {8 \over 5} x {1 \over 8} \) = \( 1 \over 5 \).
Question 3
The ratio of Iron and Zinc in an alloy is 4 : 5. In another alloy, the ratio of Iron, Copper and Zinc is 3 : 2 : 7. Equal amounts of the two alloys are molten and mixed together. What will be the ratio of Iron, Copper and Zinc in the resultant alloy?
| A. |
| B. |
| C. |
| D. |
Correct Answer : C. 25 : 6 : 41
Description :
In first alloy, ratio of iron and zinc is 4 : 5
In second alloy, ratio of iron, copper and zinc is 3 : 2 : 7
Suppose, amount T of both alloys is taken.
⇒ Amount of iron in first alloy = (4/ (4 + 5)) × T = (4/9) T
Amount of zinc in first alloy = (5/ (4 + 5)) × T = (5/9) T
Amount of iron in Second alloy = (3/ (3 + 2 + 7)) × T = (1/4) T
Amount of Copper in second alloy = (2/ (3 + 2 + 7)) × T = (1/6) T
Amount of zinc in second alloy = (7/ (3 + 2 + 7)) × T = (7/12) T
After mixing
Total amount of Iron = (4/9) T + (1/4) T = (25/36) T
Total amount of copper = (1/6) T = (6/36) T
Total amount of zinc = (5/9) T + (7/12) T = (41/36) T
⇒ Ratio of Iron, Copper and Zinc = (25/36) T : (6/36) T : (41/36) T
= 25 : 6 : 41
Question 4
Cost of two types of pulses is Rs.15 and Rs, 20 per kg, respectively. If both the pulses are mixed together in the ratio 2:3, then what should be the price of mixed variety of pulses per kg?
| A. |
| B. |
| C. |
| D. |
Correct Answer : D. Rs. 18 per kg
Description :
Let the cost of mixed variety of pulse be Rs. x
As per the alligation rule,
2:3 = (20-x) : (x-15)
⇒ 2x+3x = 60+30
⇒ 5x = 90
⇒ x = 18
Question 5
According to an instruction a mixture of colour and Turpentine containing half part of each is perfect for painting a wall. A painter is provided a mixture, 3 parts of which are colour and 5 parts are Turpentine. How much of the mixture drawn off and replaced with colour so that the mixture becomes perfect?
| A. |
| B. |
| C. |
| D. |
Correct Answer : A. 1/5th
Description :
Let total amount of the mixture be 1 litre
3 parts of the mixture are colour and 5 parts are Turpentine
⇒ Amount of colour in the mixture = 3/8 litre
⇒ Amount of turpentine in the mixture = 5/8 litre
Let amount of mixture removed by the painter be ‘x’ litre
⇒ Amount of colour in the new mixture = (3/8 – 3x/8 + x) litre
⇒ Amount of turpentine in the new mixture = (5/8 – 5x/8) litre
The mixture will be perfect if, Amount of colour in the new mixture = Amount of turpentine in the new mixture
⇒ 3/8 – 3x/8 + x = 5/8 – 5x/8
⇒ 3 – 3x + 8x = 5 – 5x
⇒ 10x = 2
⇒ x = 1/5
∴ 1/5th part of the mixture should be replaced
Question 6
A milkman claims to sell milk at its cost price, still, he is making a profit of 30% since he has mixed some amount of water in the milk. What is the % of milk in the mixture?
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| B. |
| C. |
| D. |
Correct Answer : B. 76.92 %
Description :
Let the milk he bought is 1000 ml
Let C.P of 1000 ml is Rs. 100
Here let he is mixing K ml of water
He is getting 30% profit
=> Now, the selling price is also Rs. 100 for 1000 ml
=> 100 : K%
= 100 : 30
10 : 3 is ratio of milk to water
=> Percentage of milk = 10 x 100/13 = 1000/13 = 76.92 %
Question 7
Two solutions S1 and S2 contain whisky and soda in the ratio 2 : 5 and 6 : 7 respectively. In what ratio these solutions be mixed to get a new solution S3, containing whisky and soda in the ratio 5 : 8 ?
| A. |
| B. |
| C. |
| D. |
Correct Answer : C. 7:9
Description :
Let the amount taken from S1 be 7x
And amount taken from S2 be 13y
(2x + 6y)/(5x + 7y) = 5/8
16x + 48y = 25x + 35y
9x = 13y
x/y = 13/9
Actual ratios of amounts
= 7x/13y
= (7/13) * (13/9)
= 7/9
Question 8
A container contains 40 litres of milk.From this container 4 litres of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container.
| A. |
| B. |
| C. |
| D. |
Correct Answer : C. 29.16 litres
Description :
Amount of milk left after 3 operations
\( {(40(1 − {4 \over 40})^3)} = (40 \times {9 \over 10} \times {9 \over 10} \times {9 \over 10}) \) = 29.16 litres
Question 9
A mixture of 150 liters of wine and water contains 20% water. How much more water should be added so that water becomes 25% of the new mixture?
| A. |
| B. |
| C. |
| D. |
Correct Answer : B. 10
Description :
Number of liters of water in150 liters of the mixture = 20% of 150 = 20/100 * 150 = 30 liters.
P liters of water added to the mixture to make water 25% of the new mixture.
Total amount of water becomes (30 + P) and total volume of mixture is (150 + P).
(30 + P) = 25/100 * (150 + P)
120 + 4P = 150 + P => P = 10 liters.
Question 10
The milk and water in two vessels A and B are in the ratio 4:3 and 2:3 respectively. In what ratio the liquids in both the vessels be mixed to obtain a new mixture in vessel c consisting half milk and half water?
| A. |
| B. |
| C. |
| D. |
Correct Answer : D. 7:5
Description :
Milk in 1-litre mixture of A = 4/7 litre.
Milk in 1-litre mixture of B = 2/5 litre.
Milk in 1-litre mixture of C = 1/2 litre.
By rule of alligation we have required ratio X:Y
X : Y
4/7 2/5
\ /
(Mean ratio)
(1/2)
/ \
(1/2 – 2/5) : (4/7 – 1/2)
1/10 1/1 4
So Required ratio = X : Y = 1/10 : 1/14 = 7:5