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Quantitative Aptitude

Average

Simple & Compound Interest14 / 51LCM & HCF
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Concepts (2)

Master average changes when members are added, removed, or replaced. Focus on quick calculations using total sum adjustments to ace SSC CGL questions.

Core Formulas

Understanding the effect of addition or removal on average primarily revolves around the concept of the 'sum of observations'.

  1. Basic Average Formula: Average = Sum of Observations / Number of Observations (N) Therefore, Sum of Observations = Average * N

  2. Adding a New Member: If a new member with value 'X' is added to a group of 'N' members with an average 'Avg', the new sum becomes (Avg * N) + X, and the new number of members is N+1. New Average = ((Avg * N) + X) / (N + 1) Value of Added Member (X) = (New Average * (N + 1)) - (Old Average * N)

  3. Removing a Member: If a member with value 'Y' is removed from a group of 'N' members with an average 'Avg', the new sum becomes (Avg * N) - Y, and the new number of members is N-1. New Average = ((Avg * N) - Y) / (N - 1) Value of Removed Member (Y) = (Old Average * N) - (New Average * (N - 1))

  4. Replacing a Member: If a member with value 'Y_old' is replaced by a new member with value 'Y_new' in a group of 'N' members, the number of members remains 'N'. New Sum = (Old Average * N) - Y_old + Y_new New Average = ((Old Average * N) - Y_old + Y_new) / N Change in Average = (Y_new - Y_old) / N Value of New Member (Y_new) = Value of Old Member (Y_old) + (Change in Average * N)

  5. Correcting an Incorrect Observation: Correct Sum = Incorrect Sum - Incorrect Value + Correct Value Correct Average = Correct Sum / N

Worked Example 1 (Adding a Member)

Question: The average weight of 10 students is 45 kg. If a new student joins the group, the average weight increases by 1 kg. What is the weight of the new student?

Solution:

  • Old Number of students (N) = 10
  • Old Average weight = 45 kg
  • Old Total weight = 10 * 45 = 450 kg
  • New Number of students = 10 + 1 = 11
  • New Average weight = 45 + 1 = 46 kg
  • New Total weight = 11 * 46 = 506 kg
  • Weight of new student = New Total weight - Old Total weight = 506 - 450 = 56 kg

Worked Example 2 (Removing a Member)

Question: The average age of 5 friends is 24 years. If one friend leaves, the average age of the remaining 4 friends becomes 23 years. What was the age of the friend who left?

Solution:

  • Old Number of friends (N) = 5
  • Old Average age = 24 years
  • Old Total age = 5 * 24 = 120 years
  • New Number of friends = 5 - 1 = 4
  • New Average age = 23 years
  • New Total age = 4 * 23 = 92 years
  • Age of friend who left = Old Total age - New Total age = 120 - 92 = 28 years

Shortcuts & Tricks

  1. For Added Member: If the average of N members is 'Avg' and a new member is added, increasing the average by 'x', the value of the added member is Avg + (N + 1) * x. (If average decreases by 'x', use Avg - (N + 1) * x) Example 1 Shortcut: 45 + (10 + 1) * 1 = 45 + 11 * 1 = 45 + 11 = 56 kg

  2. For Removed Member: If the average of N members is 'Avg' and a member is removed, decreasing the average by 'x', the value of the removed member is Avg + (N - 1) * x. (If average increases by 'x', use Avg - (N - 1) * x) Example 2 Shortcut: Here, avg decreased by 1. So, 24 + (5 - 1) * 1 = 24 + 4 * 1 = 24 + 4 = 28 years

  3. For Replaced Member: Value of New Member = Value of Old Member + (Change in Average * Total Members) Example: The average weight of 20 students is 30 kg. If one student (weight 25 kg) is replaced by a new student, the average increases by 0.5 kg. New student's weight = 25 + (0.5 * 20) = 25 + 10 = 35 kg.

Common Mistakes

  1. Incorrect Count (N vs N+1/N-1): Students often use the original 'N' for the new total sum calculation instead of N+1 (for addition) or N-1 (for removal).
  2. Sign Errors in Average Change: Confusing whether the average increased or decreased, leading to incorrect addition/subtraction in shortcut formulas.
  3. Mixing Up Formulas: Applying the 'added member' shortcut for a 'removed member' scenario or vice-versa.
  4. Calculation Errors: Simple arithmetic mistakes, especially when dealing with decimals or larger numbers, can lead to incorrect answers.

Derivation (brief)

Let's briefly derive the shortcut for an added member. Initial state: N members, Average = Avg. So, Sum = N * Avg. New state: N+1 members, New Average = Avg_new. Value of Added Member (X) = New Sum - Old Sum X = (N+1) * Avg_new - N * Avg If the average increases by 'x', then Avg_new = Avg + x. Substitute Avg_new: X = (N+1) * (Avg + x) - N * Avg X = N*Avg + N*x + Avg + x - N*Avg X = Avg + (N+1)*x This derivation clearly shows how the shortcut is formed, making it easier to remember and apply. Similar derivations apply for removed and replaced members.

Advanced Examples

Question: The average age of 30 students and their teacher is 15 years. If the teacher's age is excluded, the average age of the students becomes 14 years. What is the teacher's age?

Solution (Shortcut):

  • Initial group (N) = 30 students + 1 teacher = 31 members.
  • Initial Average = 15 years.
  • When teacher is excluded, N becomes 30.
  • New Average = 14 years.
  • Average decreased by 15 - 14 = 1 year.
  • This is a 'removed member' scenario where average decreased.
  • Teacher's Age = Old Average + (New Number of Members * Change in Average)
  • Teacher's Age = 15 + (30 * 1) = 15 + 30 = 45 years.

Question: The average score of a cricketer in 11 innings is 60 runs. In the 12th inning, he scores some runs, and his average increases by 2 runs. How many runs did he score in the 12th inning?

Solution (Shortcut):

  • Old Number of innings (N) = 11
  • Old Average = 60 runs
  • Average increases by 2 runs.
  • This is an 'added member' scenario where average increased.
  • Runs in 12th inning = Old Average + (New Number of Innings * Increase in Average)
  • Runs in 12th inning = 60 + (11 + 1) * 2 = 60 + 12 * 2 = 60 + 24 = 84 runs.

Variation Types

  1. Multiple Additions/Removals: Questions might involve adding or removing more than one member simultaneously or sequentially. Treat each step as a new average calculation or combine the total sum changes.
    • Example: 5 members added, total weight 200 kg. Calculate new average.
  2. Incorrect Observation/Reading: A common type where a value was misread or incorrectly entered. Use the 'Correcting an Incorrect Observation' formula.
    • Example: Average of 10 numbers was 20. Later found 35 was misread as 53. Find correct average.
  3. Group Division/Combination: Problems where a group is split or two groups are combined, affecting the average. This often uses the weighted average concept but can be solved by finding total sums.
    • Example: Average of boys is X, girls is Y. Combined average is Z. Find ratio of boys to girls.

Time-Saving Methods

  • Mental Math for Small Numbers: For simple changes, try to calculate the total sum change mentally. E.g., if 5 members average 10, total 50. Add a 6th member of 16. New total 66. New average 11. (50+16)/6 = 11.
  • Focus on the 'Difference': Instead of recalculating entire sums, focus on how much the new member's value deviates from the old average, and how that deviation is distributed among all members (N+1 or N-1).
  • Practice with Shortcuts: The more you practice the shortcut formulas, the faster you'll identify and apply them without needing to write down the full derivation.
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Start Lesson

Master Arithmetic Mean (sum/count) and Weighted Average (sum of products/sum of weights) for quick, accurate calculations in SSC CGL. Prioritize shortcuts for speed.

Core Formula

Arithmetic Mean (Simple Average) This is the most fundamental concept. It's the sum of all observations divided by the number of observations.

Average = (Sum of Observations) / (Number of Observations)

From this, we can also derive: Sum of Observations = Average × Number of Observations

Weighted Average Used when different observations or groups have different 'weights' or frequencies. The weight could be the number of items, quantity, or frequency of occurrence.

Weighted Average = (w1*x1 + w2*x2 + ... + wn*xn) / (w1 + w2 + ... + wn) Where x represents the observations/averages and w represents their respective weights.

Worked Example 1 (Arithmetic Mean)

Problem: Find the average of 15, 25, 30, and 40. Solution:

  1. Sum of observations = 15 + 25 + 30 + 40 = 110
  2. Number of observations = 4
  3. Average = 110 / 4 = 27.5

Worked Example 2 (Weighted Average)

Problem: A train travels 600 km in 5 hours and another 900 km in 10 hours. What is the average speed of the train for the entire journey? Solution: Here, the 'weights' are the time durations, and 'observations' are the speeds for each segment.

  1. Speed for first segment = 600 km / 5 hours = 120 km/hr
  2. Speed for second segment = 900 km / 10 hours = 90 km/hr
  3. Total Distance = 600 km + 900 km = 1500 km
  4. Total Time = 5 hours + 10 hours = 15 hours
  5. Average Speed = Total Distance / Total Time = 1500 km / 15 hours = 100 km/hr

Alternatively, using weighted average formula where speeds are x and times are w: Weighted Average Speed = (120*5 + 90*10) / (5 + 10) = (600 + 900) / 15 = 1500 / 15 = 100 km/hr

Shortcuts & Tricks

  1. Average Speed (Equal Distance): If a body travels from A to B at speed 'x' and returns from B to A at speed 'y' (same distance), the average speed is 2xy / (x+y). This is a specific case of weighted average where distances are equal.
    • Example (Related Exam Q1): A train travels from City A to City B at 60 km/hr and returns at 80 km/hr. Average speed = (2 * 60 * 80) / (60 + 80) = 9600 / 140 = 68.57 km/hr (approx).
  2. Deviation Method: For a series of numbers, assume an average (preferably near the middle). Calculate the deviation of each number from the assumed average. Sum the deviations and divide by the count. Add this result to the assumed average. This is faster for large numbers or many observations.

Common Mistakes

  1. Simple Average for Unequal Weights: Applying simple average (e.g., (speed1 + speed2)/2) when quantities or durations are different, instead of weighted average.
  2. Calculation Errors: Especially with large numbers or decimals, miscalculations in summing or dividing are frequent.
  3. Confusing Total vs. Average: Forgetting that Sum = Average × Count and misusing average values directly without converting to sums first.

Derivation (brief)

The concept of average is rooted in finding an 'equitable' distribution. If you have a total 'Sum' distributed among 'N' items, the average is what each item would get if the sum were divided equally. The weighted average extends this by acknowledging that not all items contribute equally to the 'Sum'. Instead of just Sum/N, it becomes (Total Weighted Sum) / (Total Weight). For instance, if you have groups of students with different average marks, the overall average isn't just the average of their group averages, but the total marks of all students divided by the total number of students. This naturally leads to (sum of (group_avg * group_size)) / (sum of group_sizes).

Advanced Examples

Problem (Related Exam Q3): The average weight of A, B, and C is 45 kg. If the average weight of A and B is 40 kg and that of B and C is 43 kg, then the weight of B is: Solution:

  1. Total weight of A, B, C = Average × Number = 45 kg × 3 = 135 kg
  2. Total weight of A and B = Average × Number = 40 kg × 2 = 80 kg
  3. Total weight of B and C = Average × Number = 43 kg × 2 = 86 kg
  4. We have: (A + B + C) = 135
  5. We have: (A + B) = 80
  6. Subtract (A+B) from (A+B+C) to find C: C = 135 - 80 = 55 kg
  7. Now use (B + C) = 86 and the value of C: B + 55 = 86 B = 86 - 55 = 31 kg Therefore, the weight of B is 31 kg.

Variation Types

  • Average Age Problems: Involving people joining or leaving a group, changing the average age.
  • Average Marks/Scores: Calculating overall average from different sections or groups of students.
  • Average Expenditure/Income: Finding the average spending or earning over periods.
  • Mixture Problems: Often solvable using the weighted average concept or Alligation, where two different concentrations/prices are mixed.

Time-Saving Methods

  1. Assumed Mean / Deviation Method: For a set of numbers (e.g., 78, 82, 75, 85, 80), assume an average (say, 80). Calculate deviations: -2, +2, -5, +5, 0. Sum of deviations = 0. So, the actual average is 80 + (0/5) = 80. If sum of deviations was +10, average would be 80 + (10/5) = 82.
  2. Alligation Rule: For weighted average problems involving two quantities (e.g., mixing two types of rice at different prices), the Alligation rule provides a quick way to find the ratio of quantities or the resultant average. If quantity Q1 with average A1 is mixed with Q2 with average A2 to get a final average A_mix, then Q1/Q2 = (A2 - A_mix) / (A_mix - A1).
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Start Lesson: Effect of Addition/Removal on Average