Mixture and Alligation is an important topic in quantitative aptitude used to solve problems related to combining two or more quantities with different values, prices, or concentrations.
Mixture
A mixture is formed by combining two or more components to obtain a final quantity with a mean value (such as average price or concentration).
These problems are generally based on the concept of ratio and proportion.
Alligation
Alligation is a method used to find the ratio in which two ingredients at different prices or concentrations must be mixed to obtain a mixture at a given mean value.
Alligation Rule:
Ratio of quantities = (Higher Value − Mean Value) : (Mean Value − Lower Value)
- Mean Price (m) is the price of one unit of the mixture and it always lies between the cheaper price (c) and the dearer price (d); in alligation, calculations are done by considering one unit only.
- In replacement questions, the quantity of pure substance left after n replacements is given by:
Quantity left = P × [1 − (R / P)]ⁿ,
where P = initial quantity, R = quantity replaced each time, and n = number of replacements.
Tips and Tricks to Solve Mixture and Alligation Questions
- The Alligation rule is a quick method to determine the ratio in which two quantities with different prices or values must be mixed to obtain a desired mean value.
- This technique helps in easily calculating the mixing ratio and cost price of the mixture.
- The concept of alligation is also useful in solving problems related to partnership, time and work, and wages, as these are based on ratio and proportion.
- Although it may appear complex initially, regular practice makes the method simple and efficient to apply.
Mixture and Alligation - Questions and Answers
Question 1: A 30-liter vessel contains pure juice. If 2 liters are taken out and replaced with water, keeping the volume constant at 30 liters, and this process is repeated 4 times, what percentage of pure juice remains in the vessel after 4 replacements?
Solution:
Here, we apply the formula (1−R/P)n
Let
P = Initial quantity of pure element = 30 liters
R = Quantity replaced each time = 2 liters
n = Number of replacements = 4So, the quantity of pure juice after 4 replacements is:
(1 − 2/30)4 = (0.9333)4 ≈ 0.7536
Therefore, the percentage of pure juice left after 4 replacements is:
0.7536 × 100 = 75.36%
Question 2 : A 60 kg alloy mixed with 90 kg of another alloy. If the first alloy contains zinc and aluminum in the ratio 4: 1, and the second alloy has aluminum and magnesium in the ratio 2:5, what is the amount of aluminum in the new alloy?
Solution:
Alloy A: 60 kg contains zinc and aluminum in the ratio 4 : 1.
So, 4 + 1 = 5
5 units → 60 kg
1 unit → 12 kg
Amount of aluminum in alloy A is 1 × 12 = 12 kg.Alloy B: 90 kg contains aluminum and magnesium in the ratio 2 : 5.
So, 2 + 5 = 7
7 units → 90 kg
1 unit → 12.857 kg
Amount of aluminum in alloy B is 2 × 12.857 ≈ 25.714 kg.Hence, the total amount of aluminum in the new alloy is 12 + 25.714 ≈ 37.7141 kg.
Question 3: 30 litres of a mixture of milk and water contains 10% of water, the water to be added, to make the water content 25% in the new mixture. Find how many litres water will be added?
Solution :
Water in the 30 litre of mixture = 30 x 10/100 = 3 litres
Milk in the mixture = 30 - 3 = 27 litres
Let x litres of water is mixed.
Acc. to question
(3 + x)/(30 + x) = 25/100
4(3 + x) = 30 + x
12 + 4x = 30 + x
3x = 18
x = 6
Hence, 6 litres of water to be added in the mixture.
Question 4: 25000 students appeared in an exam. 60% of the boys and 40% of the girls cleared the examination. If the total percent of students qualifying is 55%, how many girls appeared in the exam?
Solution :
By alligation method
4 -> 25000
1 -> 6250
Hence, number of girls passed in the exam is 6250.
Question 5: In what ratio must a shopkeeper mix sugar at Rs 30/kg and Rs 32.5/kg, so that by selling the mixture at Rs 34.1/kg he may gain 10%.
Solution :
Acc. to question
SP of 1 kg of mixture = Rs 34.1
Profit = 10% = 10/100 = 1/10
SP = 1 + 10 = 11 unit
11 units ->34.1
1 unit -> 3.1
10 unit -> 31
We obtain the CP of the mixture Rs 31.
To obtain the ratio use alligation methodHence, the ratio in which he mixed is 3:2
