Averages and Mixtures: Comprehensive Guide for CSAT

1. Introduction and Importance in CSAT

The topic of “Averages and Mixtures” (often studied alongside Alligation) is one of the foundational pillars of the quantitative aptitude section in the CSAT (UPSC/MPSC) examination. Every year, aspirants can expect 2 to 4 questions directly or indirectly based on these concepts. The beauty of Averages and Mixtures is that they are deeply interconnected with other topics such as Percentages, Profit and Loss, Time and Work, and Data Interpretation. Mastering this topic not only ensures you score marks on direct questions but also significantly boosts your calculation speed and logical deduction capabilities across the entire paper.

2. Core Concepts, Formulas, and Tricks

Averages

An average (or arithmetic mean) is a single value that represents the central tendency of a set of numbers. It is calculated by dividing the sum of all observations by the total number of observations.

• Basic Formula: Average = (Sum of all observations) / (Total number of observations)
• Sum of Observations: Sum = Average × Total number of observations
• Weighted Average: When different groups have different averages and numbers of items, the combined average is calculated as: Weighted Average = (n1 × A1 + n2 × A2 + …) / (n1 + n2 + …), where n = number of items and A = average of those items.

Shortcut Tricks for Averages:

• Deviation Method: Instead of adding large numbers, assume an average (A). Find the deviation of each number from A. The true average is A + (Sum of deviations / Total number).
• Inclusion/Exclusion: If a person joins a group and the average increases, the age/weight of the new person is: (Old Average) + (Increase in Average × New Total Number of People). If a person leaves, adjust the formula accordingly.
• Replacement: If one person is replaced by another and the average changes, the value of the new person = Value of the old person + (Change in Average × Total Number of People).

Mixtures and Alligation

A mixture is created by combining two or more different ingredients. Alligation is a rule or technique that enables us to quickly calculate the price of a mixture or the ratio in which ingredients are mixed.

• Rule of Alligation: If two ingredients A and B with unit prices C (cheaper) and D (dearer) are mixed to form a mixture with mean price M, the ratio in which they must be mixed is given by:
  Quantity of Cheaper / Quantity of Dearer = (D – M) / (M – C)
• Visualizing Alligation:
  Cheaper (C) —– Dearer (D)
  ———- Mean (M) ———
  (D – M) ——– (M – C)
• Repeated Dilution Formula: If a container initially holds ‘x’ units of pure liquid, and ‘y’ units are taken out and replaced by water, and this process is repeated ‘n’ times, the quantity of pure liquid left is:
  Final Quantity = x × (1 – y/x)ⁿ

3. Solved Examples with Step-by-Step Explanations

Example 1 (Average – Replacement)

Question: The average weight of 8 men is increased by 1.5 kg when one of the men who weighs 65 kg is replaced by a new man. The weight of the new man is?

Step-by-step Explanation:

1. The average increases by 1.5 kg for all 8 men.
2. Total increase in weight = 8 × 1.5 kg = 12 kg.
3. Since the average increased, the new man must be heavier than the man he replaced.
4. Weight of the new man = Weight of the replaced man + Total increase = 65 kg + 12 kg = 77 kg.

Answer: 77 kg

Example 2 (Average – Inclusion)

Question: The average age of a class of 39 students is 15 years. If the age of the teacher is included, then the average increases by 3 months. Find the age of the teacher.

Step-by-step Explanation:

1. Initial total age of 39 students = 39 × 15 = 585 years.
2. New total number of people = 39 + 1 (teacher) = 40.
3. New average = 15 years + 3 months = 15.25 years (since 3 months = 3/12 = 1/4 = 0.25 years).
4. New total age = 40 × 15.25 = 610 years.
5. Age of the teacher = New total age – Initial total age = 610 – 585 = 25 years.
6. Alternative Shortcut: Age of teacher = Old Average + (Increase in average × New Total number) = 15 + (0.25 × 40) = 15 + 10 = 25 years.

Answer: 25 years

Example 3 (Mixtures – Alligation)

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

Step-by-step Explanation:

1. Cost of cheaper pulse (C) = 15
2. Cost of dearer pulse (D) = 20
3. Mean price (M) = 16.50
4. Using the rule of alligation:
   Quantity of Cheaper : Quantity of Dearer = (D – M) : (M – C)
5. Ratio = (20 – 16.50) : (16.50 – 15) = 3.50 : 1.50
6. Simplify the ratio: 3.5 : 1.5 = 35 : 15 = 7 : 3.

Answer: 7:3

Example 4 (Mixtures – Repeated Dilution)

Question: 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?

Step-by-step Explanation:

1. Initial quantity of pure milk (x) = 40 litres.
2. Quantity replaced each time (y) = 4 litres.
3. Number of times the process is performed (n) = 1 (initial) + 2 (further) = 3 times.
4. Use the formula: Final Quantity = x × (1 – y/x)ⁿ
5. Final Quantity = 40 × (1 – 4/40)³ = 40 × (1 – 1/10)³
6. Final Quantity = 40 × (9/10) × (9/10) × (9/10) = 40 × 0.9 × 0.9 × 0.9
7. Final Quantity = 40 × 0.729 = 29.16 litres.

Answer: 29.16 litres

4. Pro-tips to Avoid Common Mistakes

• Unit Consistency: Always ensure that all quantities and prices are in the same units before applying the alligation formula (e.g., don’t mix grams with kilograms without converting).
• Profit and Loss in Alligation: If a mixture is sold at a profit, the selling price cannot be directly used as the mean price (M). You must first calculate the Cost Price (CP) of the mixture and use that as the mean price.
• Number of Replacements: In repeated dilution problems, pay close attention to the wording. “Repeated one more time” means n=2, while “repeated two further times” means n=3 total operations.
• Averages of Speed: Do not use the simple arithmetic mean for average speed if distances are constant. Use the formula: Average Speed = 2ab / (a + b), where a and b are speeds for equal distances.

5. Practice Questions

Test your understanding with these practice questions:

1. The average of 5 consecutive odd numbers is 61. What is the difference between the highest and lowest numbers?
2. A student’s marks were wrongly entered as 83 instead of 63. Due to that, the average marks for the class got increased by half (1/2). The number of students in the class is?
3. In what ratio must water be mixed with milk to gain 16.66% on selling the mixture at cost price?
4. A vessel contains 60 litres of a mixture of milk and water in the ratio 7:5. How many litres of water must be added to make the ratio 1:1?
5. The average temperature for Monday, Tuesday, and Wednesday was 40°C. The average for Tuesday, Wednesday, and Thursday was 41°C. If the temperature on Thursday was 42°C, what was the temperature on Monday?