Study Notes: Average (SSC MTS)

Overview

Average is a foundational arithmetic concept that appears in 3–5 questions in the SSC MTS Paper 1 exam. It tests your ability to find the central value of a group of numbers and apply this concept to real-world scenarios like age problems, marks, income, expenses, and weights.

Understanding average is critical because it connects to multiple other topics—ratio and proportion, time and speed, and even data interpretation. Questions range from straightforward computation to multi-step word problems involving weighted averages and age-related scenarios. Mastering the basic formula and its variations will give you quick marks in the exam, as these questions typically require 30–60 seconds each when you know the method.

The three main areas you must master are: (1) simple average calculation, (2) weighted averages where different groups have different counts, and (3) age-related average problems that involve addition or removal of members from a group.

Key Concepts

• **Average (Arithmetic Mean)** is the sum of all observations divided by the number of observations. It represents the central or typical value of a dataset.

• **Sum = Average × Number of observations**. This reverse formula is the most powerful tool for solving average problems quickly. Memorize it.

• When a new member joins or leaves a group, the total sum changes. Calculate the new sum, then find the new average by dividing by the new count.

• **Weighted Average** applies when different groups contribute unequally to the total. You cannot simply average the averages—you must account for the size of each group.

• In **age problems**, if n years pass, every person's age increases by n years, so the average age also increases by n years. If k people join or leave, recalculate the sum and divide by the new count.

• **Effect of adding/removing a value**: If you add a number greater than the current average, the new average increases. If you add a number less than the average, it decreases.

• When all observations increase or decrease by a constant k, the average also increases or decreases by k.

• When all observations are multiplied or divided by a constant k, the average is also multiplied or divided by k.

Formulas / Key Facts

**Average = Sum of all observations / Number of observations**

**Sum of observations = Average × Number of observations**

**New average after adding one value** = (Old sum + New value) / (Old count + 1)

**New average after removing one value** = (Old sum - Removed value) / (Old count - 1)

**Weighted Average** = (n₁A₁ + n₂A₂ + ... + nₖAₖ) / (n₁ + n₂ + ... + nₖ), where n represents counts and A represents individual averages.

**Age increase rule**: If n years pass for a group of m people, average age increases by n years (not n×m years).

**Replacement formula**: When one person is replaced, change in average = (Difference in values) / (Total number of people).

**Average of first n natural numbers** = (n + 1) / 2

**Average of first n even numbers** = (n + 1)

**Average of first n odd numbers** = n

Worked Examples

**Example 1: Basic Average Calculation**

The marks of 5 students are 45, 52, 60, 48, and 55. Find the average marks.

**Solution:** Sum = 45 + 52 + 60 + 48 + 55 = 260 Average = 260 / 5 = 52 marks

**Example 2: Finding a Missing Value**

The average of 6 numbers is 30. If one number is excluded, the average becomes 28. What is the excluded number?

**Solution:** Sum of 6 numbers = 30 × 6 = 180 Sum of 5 numbers = 28 × 5 = 140 Excluded number = 180 - 140 = 40

**Example 3: Weighted Average**

A class has 30 boys with an average weight of 60 kg and 20 girls with an average weight of 50 kg. Find the average weight of the whole class.

**Solution:** Total weight of boys = 30 × 60 = 1800 kg Total weight of girls = 20 × 50 = 1000 kg Total weight = 1800 + 1000 = 2800 kg Total students = 30 + 20 = 50 Average weight = 2800 / 50 = 56 kg

**Note:** Simply averaging 60 and 50 to get 55 kg would be WRONG because the groups are unequal.

**Example 4: Age-Related Problem**

The average age of 4 family members is 25 years. If a 5-year-old child joins the family, what is the new average age?

**Solution:** Total age of 4 members = 25 × 4 = 100 years After child joins, total age = 100 + 5 = 105 years New average = 105 / 5 = 21 years

**Example 5: Age After Time Passes**

The average age of 10 students is 20 years. What will be their average age after 5 years?

**Solution:** When 5 years pass, each student's age increases by 5 years. Therefore, average age increases by 5 years. New average = 20 + 5 = 25 years

(No need to calculate sums—use the direct rule.)

Common Mistakes

**Mistake:** Averaging the averages when groups are unequal. Students calculate (60 + 50) / 2 = 55 kg in weighted average problems. **Fix:** Always find total sum (weight × count for each group), then divide by total count. Never average averages unless all groups have equal size.

**Mistake:** In age problems, multiplying the number of years by the number of people. If 5 years pass for 10 people, students wrongly add 50 years to total age. **Fix:** When time passes equally for everyone, the average age simply increases by that number of years. Each person ages by n years, so average also increases by n years.

**Mistake:** Forgetting to adjust the count when someone joins or leaves. Students divide by the old count after adding a new member. **Fix:** Always update both numerator (sum) and denominator (count). New average = New sum / New count.

**Mistake:** Confusing "average increased by 2" with "total increased by 2." If average of 5 numbers increases by 2, the total increases by 10. **Fix:** Change in sum = Change in average × Number of observations. Remember the fundamental relationship.

**Mistake:** Not using the reverse formula efficiently. Students add all numbers when the average and count are already given. **Fix:** Master "Sum = Average × Count" and use it immediately when you see average and count together. This saves calculation time and reduces errors.

Quick Reference

• Average = Sum / Count; Sum = Average × Count (use this reverse formula constantly)
• Weighted average requires summing (group size × group average) for all groups, then dividing by total count
• When time passes equally for everyone, average age increases by exactly that time period
• When one value is added: new sum = old sum + new value; new count = old count + 1
• Change in total sum = Change in average × Number of observations
• Cannot average averages unless all groups have equal size—always go back to total sum