5. Average and weighted average
Free study material · concepts, shortcuts & solved questions
Average = total ÷ number of items. If the average of n values changes by k, the total changes by nk. A combined average must weight each group average by its group size; never average two averages unless group sizes are equal.
Method before formula Use assumed average for close values: average = assumed value + (sum of deviations ÷ n). In replacement questions, change in average = change in total ÷ number of items. |
Formula and decision table
Situation | Fast relation |
Combined average | (n₁a₁+n₂a₂)/(n₁+n₂) |
Replacement | New total = old total − old item + new item |
Consecutive numbers | Average is the middle number |
Worked understanding
A reliable solution has five visible moves: identify the data, choose the base or unit, write the rule, substitute carefully, and check the size of the answer. The examples in this chapter are deliberately small so that the method remains visible.
Calculation discipline Before pressing ahead, state the denominator or unit in words. For example: profit on CP, discount on MP, average over number of items, speed in metres per second, and probability over the fixed sample space. |
Solved examples from this chapter
Question focus | Correct move | Answer |
Average of 8 numbers is 24. Sum: | Sum=8×24=192. | B. 192 |
Groups of 5 and 3 have averages 42 and 58. Combined average: | (5×42+3×58)/8=384/8=48. The group sizes must weight the two averages. | B. 48 |
Average of 7 consecutive integers is 31. Largest: | The middle is 31; numbers run 28 to 34. | B. 34 |
What an examiner is testing
The options around Average and weighted average usually represent a wrong base, a missed conversion, a sign error or a shortcut used outside its condition. Say the base and unit before calculating, estimate the result, and use substitution or a reverse operation as the final check.
Step-by-step answer construction
Example 1: Average of 8 numbers is 24. Sum: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 192, because Sum=8×24=192.
Example 2: Groups of 5 and 3 have averages 42 and 58. Combined average: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 48, because (5×42+3×58)/8=384/8=48. The group sizes must weight the two averages.
Example 3: Average of 7 consecutive integers is 31. Largest: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 34, because The middle is 31; numbers run 28 to 34.
Chapter practice
1. Average of 8 numbers is 24. Sum:
(A) 32 (B) 192 (C) 216 (D) 240
Answer: B. 192 | Explanation: Sum=8×24=192.
2. Groups of 5 and 3 have averages 42 and 58. Combined average:
(A) 46 (B) 48 (C) 48.5 (D) 50
Answer: B. 48 | Explanation: (5×42+3×58)/8=384/8=48. The group sizes must weight the two averages.
3. Average of 7 consecutive integers is 31. Largest:
(A) 33 (B) 34 (C) 35 (D) 37
Answer: B. 34 | Explanation: The middle is 31; numbers run 28 to 34.
SSC CGL speed and trap clinic
For Average and weighted average, speed comes after classification. Before calculating, say aloud what the number means, what the denominator is, and what unit the answer must carry. Then use the nearest-option check to catch a sign, base or conversion error.
Checkpoint | What to verify | Typical SSC mistake |
Base | Which quantity is the base in Average and weighted average? | Using selling price instead of cost price, or part instead of total |
Unit | Are time, distance, area and rate in compatible units? | Mixing hours with minutes or km/h with m/s |
Direction | Should the answer increase, decrease or stay bounded? | Accepting an impossible negative length or probability above 1 |
Estimate | What range should the answer lie in before exact work? | Trusting a long calculation that is far from the options |
Reverse check | Can the answer be substituted back into the condition? | Stopping at an algebraic value without verification |
Mini decision drill
1. Write the first operation you would perform in a Average and weighted average question and why.
2. State the most dangerous denominator or unit in this chapter.
3. Give one condition under which a shortcut would be invalid.
4. Create a small numerical example and verify it by a second method.
Revision evidence Do not tick this chapter because you recognised the formula. Tick it only after you solve one direct question, one altered-condition question and one mixed-paper question correctly. |