Solved problems of Mixture & Alligation :

Question 1  किस अनुपात में, फलों के रस के साथ जिसका मूल्य 24 रुपये प्रति लीटर है में पानी को मिलाया जाना चाहिए ताकि रस का मूल्य 20 रुपये प्रति लीटर हो जाये ?

a) 1:4

b) 1:5

c) 1:6

d) 2:5

Answer : b) 1:5

Solution :

Cost of 1 litre of water = Rs.0 = cheaper quantity.
Cost of 1 litre of juice = Rs.24 = dearer quantity.
And, the mean price = m = Rs.20
Applying the rule of alligation,

Therefore, (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c) = 4:20 = 1:5
Hence, the required answer is 1:5.

Question 2  बॉक्स A में 30 रुपये प्रति किलोग्राम मूल्य का गेहूं  है और बॉक्स B में   40 रुपये प्रति किलोग्राम की कीमत वाला गेहूं है यदि A और B दोनों को अनुपात 4: 7 में मिश्रित किया जाता है  तो प्रति किग्रा के मिश्रण की कीमत क्या होगी? :

a) Rs. 36.36

b) Rs. 35.80

c) Rs. 42.50

d) 31. 30

Answer : a) Rs. 36.36

Solution :

Cost of 1 kg of A = Rs.30 = cheaper quantity.
Cost of 1 kg of B = Rs.40 = dearer quantity.
Let the mean price be Rs.X.
Applying the rule of allegation,

Therefore, (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c) = (40-X) : (X-30)
Given ratio = 4/7 = 40-X/X-30
280-7X = 4X - 120
11X = 400
X = 400/11 = 36.36
Hence, the required mean price is Rs.36.36

Question 3  एक विक्रेता के पास दो किस्म के चावल हैं जो कि 80 रुपये प्रति किग्रा और 9 5 रुपये किलो प्रति किलो हैं। अगर दोनो किस्मों के मिश्रण को 84 रुपये प्रति किलो में बेचता है तो 20% लाभ मिलता है तो मिश्रण अनुपात है:

a) 7:5

b) 5:3

c) 7:2

d) 5:2

Answer : d) 5:2

Solution :

Cost of 1 kg of 1st type = Rs.80 = cheaper quantity.
Cost of 1 kg of 2nd type = Rs.95 = dearer quantity.
Selling price of 1 kg of mixture = Rs. 84
Given that, gain = 20%
We have to find the cost price of 1 kg of mixture.

Selling price cost price
  Rs.120    Rs.100
  Rs. 1    Rs. 100/120
  Rs.84    Rs.100 x 84 / 120  = Rs.70.

Therefore, mean price of 1 kg of mixture = Rs.70

Therefore, (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c) = 25:10 = 5:2
Hence, required ratio is 5:2.

Question 4  दो किस्म की चाय की कीमत 90 रुपये प्रति किग्रा और 70 रुपये प्रति किलो है। यदि X किलोग्राम 1 प्रकार की छाए को 56 किलोग्राम दूसरे प्रकार की छाए के साथ मिलाया जाता है तो मिश्रण को 95 रुपये प्रति किलोग्राम पर बेचकर लाभ 25% मिलता है तो X बराबर है:

a) 35

b) 24

c) 48

d) 19

Answer : b) 24

Solution :

Cost of 1 kg of 2nd kind = Rs.70 = cheaper quantity.
Cost of 1 kg of 1st kind = Rs.90 = dearer quantity.
Selling price of 1 kg of mixture = Rs. 95
Gain = 25%
Therefore cost price of 1 kg of mixture = 100/125 x 95 = Rs.76
That is, mean price = m = Rs.76

Therefore, (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c) = 14:6 = 7:3.
Ratio of quantities of 2nd and 1st kind of tea = 7:3
Given that, 56kg of 2nd is mixed with X kg of 1st.
Then, 7:3 = 56:X
7X = 56x3
X = 24kg
Hence, required answer is 24.