# Alligation or Mixture Questions

FACTS  AND  FORMULAE  FOR  ALLIGATION  OR  MIXTURE  QUESTIONS

I. 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 a desired price.

II. Mean Price : The cost price of a unit quantity of the mixture is called the mean price.

III. Rule of Alligation : Suppose Rs.x per unit be the price of first ingradient mixed with another ingradient (cheaper) of price Rs.y per unit to form a mixture whose mean price is Rs. z per unit, then

Quantity of cheaper : Quantity of dearer

= ( C.P of dearer - Mean Price ) : ( Mean Price - C. P of cheaper )

= ( x- z ) : ( z - y )

IV. Suppose a container contains x units of liquid from which y units are taken out and replaced by water. After  operations , the quantity of pure liquid = $x{\left(1-\frac{y}{x}\right)}^{n}$ units.

Q:

Milk and water are mixed in a vessel A as 4:1 and in vessel B as 3:2. For vessel C, if one takes equal quantities from A and B, find the ratio of milk to water in C.

 A) 1:9 B) 9:1 C) 3:7 D) 7:3

Explanation:

Ratio of Milk and water in a vessel A is 4 : 1

Ratio of Milk and water in a vessel B is 3 : 2

Ratio of only milk in vessel A = 4 : 5

Ratio of only milk in vessel B = 3 : 5

Let 'x' be the quantity of milk in vessel C

Now as equal quantities are taken out from both vessels A & B

=> 4/5     :     3/5
x
3/5-x          x - 4/5

=> $35-xx-45$ = $11$ (equal quantities)

=> x = 7/10

Therefore, quantity of milk in vessel C  = 7

=> Water quantity = 10 - 7 = 3

Hence the ratio of milk & water in vessel 3 is 7 : 3

33 16254
Q:

How many liters of oil at Rs.40 per liter should be mixed with 240 liters of a second variety of oil at Rs.60 per liter so as to get a maximum whose cost is Rs.52 per liter ?

Apply Allegation Method and first calculate the ratio in which they have to be mixed.

= 8 : 12 = 2 : 3

Thus, the two varieties of oil should be mixed in the ratio 2 : 3. So, if 240 liters of the second variety are taken, then 160 liters of the first variety should be taken.

15871
Q:

4 kg of a metal contains 1/5 copper and rest in Zinc. Another 5 kg of metal contains 1/6 copper and rest in Zinc.The ratio of Copper and Zinc  into the mixture of these two metals:

 A) 49 : 221 B) 39:231 C) 94:181 D) None of these

Explanation:

Copper in 4 kg = 4/5 kg          and      Zinc in 4 kg = 4 x (4/5) kg

Copper in 5 kg = 5/6 kg          and      Zinc in 5 kg = 5 x (5/6) kg

Therefore, Copper in mixture = $45+56=4930$ kg

and  Zinc in the mixture = $165+256=22130$kg

Therefore the required ratio = 49 : 221

30 15747
Q:

From the 50 liters of milk, 5 liters of milk is taken out and after it 5 liters of water is added to the rest amount of milk. Again 5 liters of milk and water is drawn out and it was replaced by 5 liters of water. If this process is continued  similarly for the third time, the amount of milk left after the third replacement:

 A) 45L B) 36.45L C) 40.5L D) 42.5L

Explanation:

General Formula:

Final or reduced concentration = initial concentration x

where n is the number of times the same operation is being repeated. The "amount being replaced" could be pure or mixture as per the case. similarly ,"total amount" could also be either pure or mixture. Here amount being replaced denotes the quantity which is to be withdrawn in each time.

Therefore,    $50×1-5503$

= 36.45 L

20 15743
Q:

The amount of water (in ml) that should be added to reduce 9 ml lotion, containing 50% alcohol, to a lotion containing 30% alcohol is ?

 A) 6 ml B) 11 ml C) 15 ml D) 9 ml

Explanation:

Let us assume that the lotion has 50% alcohol and 50% water.
ratio = 1:1
As the total solution is 9ml
alcohol = water = 4.5ml
Now if we want the quantity of alcohol = 30%
The quantity of water = 70%
The new ratio = 3:7
Let x ml of water be added
We get,

=> x=6
Hence 6ml of water is added.

26 15254
Q:

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?

 A) 71.02% B) 76.92% C) 63.22% D) 86.42%

Explanation:

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%

26 14144
Q:

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) 5 lit B) 10 lit C) 15 lit D) 20 lit

Explanation:

Number of liters of water in 150 liters of the mixture = 20% of 150 = 20/100 x 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 x (150 + P)
120 + 4P = 150 + P => P = 10 liters.

9 13792
Q:

A mixture of 70 litres of Fruit Juice and water contains 10% water. How many litres of water should be added to the mixture so that the mixture contains 12.5% water   ?

 A) 2 lit B) 4 lit C) 1 lit D) 3 lit

Explanation:

Quantity of fruit juice in the mixture = 70 - [70 x (10/100) ]= 63 litres.

After adding water, juice would form 87.5% of the mixture.

Hence, if quantity of mixture after adding x liters of water, [(87.5) /100 ]*x = 63 => x = 72

Hence 72 - 70 = 2 litres of water must be added.