Weighted Average: Combined Groups and Average Speed
Weighted Average: Combined Groups and Average Speed
When groups of different sizes are combined, the overall average is NOT the simple average of the group averages; each group counts in proportion to its size. The same idea gives the ratio of group sizes from averages, and average speed over parts of a journey.
1. Formula Box
Weighted average = (n₁A₁ + n₂A₂ + ... + nₖAₖ)/(n₁ + n₂ + ... + nₖ)
Two groups with averages A₁ < A < A₂ (A = combined): n₁ : n₂ = (A₂ − A) : (A − A₁)
Average speed = total distance / total time
Equal distances at speeds x and y: average speed = 2xy/(x + y)
Equal times at speeds x and y: average speed = (x + y)/2
| Data | Combined average | Note |
|---|---|---|
| 40 at 62, 35 at 72 | 5,000/75 = 66.67 | Closer to 62 because that group is bigger |
| Equal distances at 40 and 60 km/h | 2 × 40 × 60/100 = 48 km/h | Not 50 |
| Equal times at 40 and 60 km/h | 50 km/h | Simple average works only for equal times |
The ratio rule is the alligation rule (next topic): the bigger group pulls the combined average towards itself, and the distances from the mean are in the inverse ratio of the group sizes.
✗ Averages 62 and 72, so the combined average is 67 | ✓ Only if the groups are equal; weight each average by its group size
✗ Average speed for 40 and 60 km/h = 50 km/h | ✓ For equal distances use 2xy/(x + y) = 48 km/h
हिंदी नोट: भारित औसत में हर समूह का औसत उसकी संख्या से गुणा होता है। समान दूरी अलग-अलग चाल से तय करने पर औसत चाल 2xy/(x + y) होती है, साधारण औसत नहीं।
Exam Pointer: Verified NTPC patterns: average speed over a journey in parts (January 2021, August 2025 UG CBT-1). Combined averages and the ratio of group sizes had no verified NTPC shift in our check and are tagged syllabus-based; they are the base for alligation and for combined-marks questions in data interpretation.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Combined average of two or more groups
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Section A has 40 students with average marks 62 and section B has 35 students with average marks 72. Find the average of both sections together.
Totals: 40 × 62 = 2,480 and 35 × 72 = 2,520. Combined = 5,000/75 = 66.67 (66 2/3).
Answer: 66 2/3
EXAMATLAS LEVEL
Q. Three sections have 30, 40 and 50 students with average marks 62, 55 and 52. Find the overall average.
Total = 1,860 + 2,200 + 2,600 = 6,660 over 120 students: 6,660/120 = 55.5. The simple average of 62, 55, 52 is 56.33, the trap option.
Answer: 55.5
Pattern 2: Ratio of group sizes from the averages
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The average marks of boys in a class are 70 and of girls 80. The class average is 74. Find the ratio of boys to girls.
Boys : girls = (80 − 74) : (74 − 70) = 6 : 4 = 3 : 2.
Answer: 3 : 2
EXAMATLAS LEVEL
Q. The average salary of 60 employees is ₹25,000. Officers earn ₹45,000 on average and the others ₹20,000. How many officers are there?
Officers : others = (25,000 − 20,000) : (45,000 − 25,000) = 5,000 : 20,000 = 1 : 4. Officers = 60/5 = 12. Check: 12 × 45,000 + 48 × 20,000 = 15,00,000 = 60 × 25,000.
Answer: 12
Pattern 3: Average speed for a journey in parts
[PYQ: NTPC UG CBT-1 13-Aug-2025 Shift-1 | NTPC CBT-1 19-Jan-2021 Shift-1]
EXAM LEVEL
Q. A man goes from A to B at 40 km/h and returns at 60 km/h. Find his average speed for the whole journey.
Equal distances: 2 × 40 × 60/(40 + 60) = 4,800/100 = 48 km/h.
Answer: 48 km/h
EXAMATLAS LEVEL
Q. A car travels 120 km at 40 km/h, then 150 km at 50 km/h, then 90 km at 45 km/h. Find its average speed.
Times: 3 h, 3 h and 2 h, total 8 h. Distance 360 km. Average speed = 360/8 = 45 km/h. Averaging 40, 50 and 45 also gives 45 here only by coincidence; always use total distance ÷ total time.
Answer: 45 km/h
Pattern 4: Overall percentage across papers of different totals
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A student scores 60% in a paper of 150 marks and 80% in a paper of 100 marks. Find his overall percentage.
Marks: 90 + 80 = 170 out of 250, which is 68%. The simple average of 60% and 80% (70%) ignores that the 150-mark paper weighs more.
Answer: 68%
EXAMATLAS LEVEL
Q. To pass, a student needs 65% overall in three papers of 100, 150 and 250 marks. He scores 58% and 72% in the first two. What minimum percentage must he score in the third paper?
Required total = 65% of 500 = 325. He has 58 + 108 = 166, so he needs 159 out of 250 = 63.6% in the third paper.
Answer: 63.6%
60-Second Revision
- Weighted average = Σ(size × average) ÷ Σ size.
- Group ratio = (higher − mean) : (mean − lower), inverse distances.
- Average speed = total distance ÷ total time; equal distances 2xy/(x + y); equal times (x + y)/2.
- Overall percentage = total marks scored ÷ total maximum marks, never the average of percentages.
Next Step: Next, the Mixture and Alligation page uses this same ratio rule for prices, mixtures and percentages.