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UGC NET PAPER 1 · UNIT 05 · MODULE 5: AVERAGES, WORK AND MOTION · TOPIC 12

Average and Ages

Average formulas, replacing and adding items, and the standard age-problem set-ups.

  • 7 sub-topics
  • 5 min read
  • Topic 12 of 17

What this topic covers

7 sub-topics, in the order to study them. Tap one to jump to it.

  1. §OverviewThe average (arithmetic mean) is the sum of all observations divided by the number of observations.
  2. 12.1Core formulasConcept Formula Average Sum of observations / number of observations Sum Average x number of observations Average of first n natural numbers (n + 1)/2 Average of first n…
  3. 12.2PropertiesThe average always lies between the smallest and the largest value. If every item rises by k, the average rises by k. If every item is multiplied by k, the average is…
  4. 12.3Worked examples (averages)Example 1.
    • Example 1
    • Example 2
    • Example 3
    • Example 4
    • Example 5
  5. 12.4Ages: key ideasCovers difference.
    • difference
  6. 12.5Worked examples (ages)Example 8.
    • Example 8
    • 12 years
    • 20 years
    • Example 9
    • 10 years
  7. 12.6TricksCovers Assumed mean (deviation method), Use the change-in-total idea, Use equal steps for consecutive numbers, Age gap is fixed.
    • Use the change-in-total idea
    • Age gap is fixed
  8. 12.7TrapsAveraging speeds directly when distances are equal (use 2ab/(a + b)). Using the old count of items when an item is added or removed. Treating a ratio of ages as the ages…

Revision mode hides the full explanations and keeps each section's key terms, lists and tables. Use it from your second read onward.

§Overview#

The average (arithmetic mean) is the sum of all observations divided by the number of observations. Average questions almost always reduce to one idea: sum = average x number of items. Age problems use the same idea along with the fact that the gap between two ages never changes.

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12.1Core formulas#

In brief

Concept Formula Average Sum of observations / number of observations Sum Average x number of observations Average of first n natural numbers (n + 1)/2 Average of first n…

Concept Formula
Average Sum of observations / number of observations
Sum Average x number of observations
Average of first n natural numbers (n + 1)/2
Average of first n even numbers n + 1
Average of first n odd numbers n
Average of squares of first n natural numbers (n + 1)(2n + 1)/6
Average of an arithmetic progression (first term + last term)/2
Average of consecutive numbers The middle number (for an odd count) or the mean of the two middle numbers (for an even count)
Combined average of two groups (n1 a1 + n2 a2)/(n1 + n2)
Weighted average Sum of (weight x value)/sum of weights
Average speed for equal distances at speeds a and b 2ab/(a + b)
Average speed for equal times at speeds a and b (a + b)/2
A person replaced; average changes by d over n persons New person = old person + n x d
A new item added; average changes by d for n old items New item = old average + (n + 1) x d
Average age after t years Old average + t (same group)

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12.2Properties#

In brief

The average always lies between the smallest and the largest value. If every item rises by k, the average rises by k. If every item is multiplied by k, the average is…

  • The average always lies between the smallest and the largest value.

  • If every item rises by k, the average rises by k. If every item is multiplied by k, the average is multiplied by k.

  • The sum of deviations from the average is always zero.

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12.3Worked examples (averages)#

In brief

Example 1.

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • 48 km/h

Example 1. The average of 5 numbers is 27. One number is excluded and the average of the rest becomes 25. Find the excluded number.

  • Total = 5 x 27 = 135. New total = 4 x 25 = 100.

  • Excluded number = 135 - 100 = 35.

Example 2. 30 students have an average mark of 40 and 20 students have an average mark of 60. Find the combined average.

  • (30 x 40 + 20 x 60)/50 = (1200 + 1200)/50 = 48.

Example 3. Ravi's average over 5 tests is 72, and after the 6th test it is 75. Find his 6th score.

  • Total after 6 tests = 6 x 75 = 450. Total after 5 tests = 5 x 72 = 360.

  • 6th score = 90.

Example 4. The average of 10 numbers is 15. One number was read as 26 instead of 36. Find the correct average.

  • Correct sum = 150 + 10 = 160. Correct average = 16.

Example 5. The average of 8 persons rises by 2 when one person of weight 65 is replaced by a new person. Find the new person's weight.

  • New person = 65 + 8 x 2 = 81.

Example 6. The average of 10 numbers is 20. Another number is added and the average becomes 21. Find the new number.

  • New number = 20 + 11 x 1 = 31. (Check: (200 + 31)/11 = 21.)

Example 7. A car goes uphill at 40 km/h and returns downhill over the same distance at 60 km/h. Find the average speed.

  • 2 x 40 x 60/(40 + 60) = 48 km/h.

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12.4Ages: key ideas#

In brief

Covers difference.

  • difference

  • The difference between two people's ages stays the same for ever.

  • After t years every person's age increases by t, so the total for n people rises by n x t.

  • Express present ages as ratio parts (3k and 5k), form an equation with the condition given, and solve for k.

  • A person's age t years ago is present age - t. A person who is not yet born (for example a newborn) has age 0 at birth.

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12.5Worked examples (ages)#

In brief

Example 8.

  • Example 8
  • 12 years
  • 20 years
  • Example 9
  • 10 years
  • 40 years
  • Example 10
  • 28 years

Example 8. The present ages of A and B are in the ratio 3 : 5. After 6 years the ratio becomes 9 : 13. Find their present ages.

  • (3k + 6)/(5k + 6) = 9/13, so 39k + 78 = 45k + 54, which gives 6k = 24 and k = 4.

  • A = 12 years, B = 20 years. (After 6 years: 18 and 26, which is 9 : 13.)

Example 9. A father is four times as old as his son. After 5 years he will be three times as old. Find their present ages.

  • Let the son be x and the father 4x. Then 4x + 5 = 3(x + 5), so x = 10.

  • Son = 10 years, father = 40 years.

Example 10. The average age of 5 family members is 24 years. The youngest, aged 8, leaves. Find the average age of the remaining four.

  • Total = 120. After the youngest leaves: 112. Average = 28 years.

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12.6Tricks#

In brief

Covers Assumed mean (deviation method), Use the change-in-total idea, Use equal steps for consecutive numbers, Age gap is fixed.

  • Use the change-in-total idea
  • Age gap is fixed

  • Assumed mean (deviation method): pick a convenient value near the data, add the deviations, divide by the count and adjust. Marks 78, 82, 85, 79, 76 with an assumed mean of 80 have deviations -2, +2, +5, -1, -4 (sum 0), so the average is exactly 80.

  • Use the change-in-total idea: the change in total = (number of items) x (change in average).

  • Use equal steps for consecutive numbers: the middle number is the average, so five consecutive numbers with an average of 15 are 13, 14, 15, 16, 17.

  • Age gap is fixed: use the gap, which does not change, to simplify problems.

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12.7Traps#

In brief

Averaging speeds directly when distances are equal (use 2ab/(a + b)). Using the old count of items when an item is added or removed. Treating a ratio of ages as the ages…

  • Averaging speeds directly when distances are equal (use 2ab/(a + b)).

  • Using the old count of items when an item is added or removed.

  • Treating a ratio of ages as the ages themselves.

  • Forgetting to add the years to every person when ages are projected into the future.

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Test yourself on Average and Ages

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↑ Module 5: Averages, Work and Motion · All 18 topics in Mathematical Reasoning · Unit overview and sample MCQs

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