Quants 6 - Alligations

Key Points to Remember:

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 the mixture of a desired price.
Mean Price -  The cost price of a unit quantity of the mixture is called the mean price
Rule of Alligation - If two ingredients are mixed, then
Quality of Cheaper / Quality of Dearer = [ CP of dearer - Mean Price ] / [ Mean Price - CP of cheaper ]

Practice Questions:

Question 1:

In what ratio must rice at ₹9.30 per kg be mixed with rice at ₹10.80 per kg so that the mixture be worth ₹10 per kg?

Cost Price of 1st rice in paise (c) = 930
Cost Price of 2nd rice in paise (d) = 1080
Mean price in paise (m) = 1000
Cheaper quantity (d-m) = 1080 - 1000 = 80
Dearer quantity (m-c) = 1000 - 930 = 70
Required Ratio = 80 : 70 = 8 : 7

Question 2:

In what ratio must water be mixed with milk to gain 20% by selling the mixture at cost price

Let the cost price of milk be ₹1 per litre.
Then selling price of 1 litre of mixture be ₹1.
Gain obtained = 20%

Cost price of 1 litre of mixture = ₹ (100/120) * 1 = 5/6

Cost Price of water (c) = 0
Cost Price of milk (d) = 1
Mean price (m) = 5/6
Cheaper quantity (d-m) = 1 - 5/6 = 1/6
Dearer quantity (m-c) = 5/6 - 0 = 5/6
Required Ratio of Water and Milk = 1/6 : 5/6 = 1 : 5

Question 3:
In what ratio must a grocer mix two varieties of pulses costing ₹15 and ₹20 per kg respectively so as to get a mixture worth ₹16.50 per kg?

Cost Price of 1st pulse (c) = 15
Cost Price of 2nd pulse (d) = 20
Mean price (m) = 16.50
Cheaper quantity (d-m) = 20 - 16.50 = 3.50
Dearer quantity (m-c) = 16.50 - 15 = 1.50
Required Ratio = 3.50 : 7.50 = 7 : 3 


Question 4:
Find the ratio in which rice at ₹7.20 a kg be mixed with rice at ₹5.70 a kg to produce a mixture worth ₹6.30 a kg.

Cost Price of 1st rice in paise (c) = 720
Cost Price of 2nd rice in paise (d) = 570
Mean price in paise (m) = 630
Quantity 1 (d-m) = 720 - 630 = 90
Quantity 2  (m-c) = 630 - 570 = 60
Required Ratio = 60 : 90 = 2 : 3


Question 5:

In what ratio must tea at ₹62 per kg be mixed with tea at ₹72 per kg so that the mixture must be worth ₹64.50 per kg?

Cost Price of 1st tea (c) = 62
Cost Price of 2nd tea (d) = 72
Mean price (m) = 64.50
Cheaper quantity (d-m) = 72 - 64.50 = 7.50
Dearer quantity (m-c) = 64.50 - 62 = 2.50
Required Ratio = 7.50 : 2.50 = 3 : 1 


Time to Think:

In what ratio must water be mixed with milk costing ₹12 per litre to obtain a mixture worth of ₹8 per litre?