1. Chapter at a Glance
Section A of a class has 20 students with an average of 80 marks; Section B has 40 students averaging 65. Is the average of the whole class (80 + 65) ÷ 2 = 72.5? No! Section B has twice as many students, so it should "count" twice as much. The true class average is a weighted average. Weighted averages appear whenever quantities combine — mixing solutions, calculating final marks from tests of different importance, product ratings and more. This chapter also introduces stacked and 100% stacked bar charts, and what each kind of chart shows or hides.
- Simple average (mean) vs weighted average.
- Combining averages of groups of different sizes.
- Weighted marks and rating schemes.
- Mixtures: combining prices, concentrations and speeds.
- Stacked bar charts (totals) vs 100% stacked bar charts (proportions).
- What inferences each chart allows — and what it can hide.
Note: all data in this chapter is practice data for learning.
2. Simple Average
The simple average (mean) treats every value equally: mean = sum ÷ count. It is right when every value represents the same "amount" — one student, one day, one item.
3. Weighted Average
If values x₁, x₂, …, xₙ have weights w₁, w₂, …, wₙ, then
Weighted average = (w₁x₁ + w₂x₂ + … + wₙxₙ) ÷ (w₁ + w₂ + … + wₙ)
Section A: 20 students, mean 80. Section B: 40 students, mean 65. Find the class mean.
Solution: Total marks = 20 × 80 + 40 × 65 = 1600 + 2600 = 4200. Class mean = 4200 ÷ 60 = 70 (not 72.5).
- The weighted average always lies between the smallest and largest values.
- It is closer to the value with the bigger weight.
- If all weights are equal, it equals the simple average.
- For two values, the distances from the average are in the inverse ratio of the weights: (70 − 65) : (80 − 70) = 5 : 10 = 1 : 2, while weights are 40 : 20 = 2 : 1.
4. Weighted Marks and Ratings
A school computes final marks as: projects 20%, mid-term 30%, final exam 50%. Arjun scores 90, 70 and 76. Find his final mark.
Solution: 0.2 × 90 + 0.3 × 70 + 0.5 × 76 = 18 + 21 + 38 = 77.
A product received: 5 stars × 40 reviews, 4 stars × 30, 3 stars × 10, 2 stars × 5, 1 star × 15. Find the average rating.
Solution: (200 + 120 + 30 + 10 + 15) ÷ 100 = 375 ÷ 100 = 3.75 stars.
5. Mixtures
30 kg of rice at ₹40/kg is mixed with 20 kg at ₹55/kg. Find the price of the mixture per kg.
Solution: (30 × 40 + 20 × 55) ÷ 50 = (1200 + 1100) ÷ 50 = ₹46/kg.
200 mL of 10% salt solution is mixed with 300 mL of 20% salt solution. Find the concentration of the mixture.
Solution: Salt = 20 + 60 = 80 mL in 500 mL → 16%.
A bus goes 60 km at 30 km/h and returns at 60 km/h. Find the average speed.
Solution: Time = 2 h + 1 h = 3 h; distance = 120 km → 40 km/h (not 45). Speeds must be weighted by time, not by distance.
Finding a ratio (alligation idea): To mix ₹40 and ₹55 rice to get ₹46 rice, the quantities must be in ratio (55 − 46) : (46 − 40) = 9 : 6 = 3 : 2.
6. Stacked Bar Charts
A stacked bar chart places the parts of a total one on top of another in a single bar. Its height shows the total, and each segment shows a part.
Practice data — students choosing activities in two schools:
| School | Sports | Music | Art | Total |
|---|---|---|---|---|
| School P | 120 | 60 | 20 | 200 |
| School Q | 150 | 150 | 100 | 400 |
7. 100% Stacked Bar Charts
A 100% stacked bar converts each part to a percentage of its own total, so every bar has the same height (100%). It compares proportions, not sizes.
| Chart | Good for | Hides |
|---|---|---|
| Stacked bar | Comparing totals; seeing the biggest parts | Proportions when totals differ |
| 100% stacked bar | Comparing shares/percentages | The actual sizes of groups |
From the 100% chart you cannot say "more students play sports in P than in Q" — in fact Q has 150 and P has 120. You can only say "a larger fraction of P's students choose sports".
8. Solved Problems (Board Pattern)
The mean of 30 boys is 60 kg and the mean of the whole class is 55 kg. If the girls' mean is 50 kg, how many girls are there?
Solution: 30 × 60 + 50g = 55(30 + g) → 1800 + 50g = 1650 + 55g → g = 30.
Using the schools data, find the percentage of art students in each school.
Solution: P: 20/200 = 10%; Q: 100/400 = 25%.
In town X, 40% of 5,000 people use buses; in town Y, 60% of 15,000. What percent use buses overall?
Solution: (2000 + 9000) ÷ 20000 = 55% (not the simple average 50%).
9. Master Formula Table
| Idea | Formula |
|---|---|
| Simple mean | Σx ÷ n |
| Weighted mean | Σ(w·x) ÷ Σw |
| Combined mean of two groups | (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂) |
| Mixture price | (q₁p₁ + q₂p₂) ÷ (q₁ + q₂) |
| Mixing ratio (alligation) | q₁ : q₂ = (p₂ − m) : (m − p₁) |
| Average speed | Total distance ÷ total time |
| Same distance at u and v | 2uv ÷ (u + v) |
| Share in 100% bar | part ÷ total × 100 |
10. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Weighted average | Bigger weight pulls harder | Combine group means |
| Marks schemes | Percent weights add to 100% | Final grade calculations |
| Mixtures | Weight by quantity | Price/concentration of mix |
| Average speed | Weight by time | Round trips |
| Stacked bar | Totals and parts | Read totals |
| 100% stacked bar | Proportions only | Valid and invalid inferences |
11. Mind Map
- Simple
- Weighted
- Class means, grades
- Ratings
- Price, concentration
- Speed by time
- Stacked = totals
- 100% = shares
12. Tips and Tricks
- Convert every average back to a total (mean × count), add totals, then divide.
- Quick check: the answer must lie between the group values, nearer the heavier group.
- Before reading a chart, ask: does it show counts or percentages?
- Averaging averages without weights.
- Averaging speeds directly for equal distances.
- Comparing sizes using a 100% stacked bar chart.
13. Important One-liners
- A weighted average gives more importance to values with larger weights.
- Combined mean = total of all values ÷ total count.
- A weighted average lies between the smallest and largest values.
- Average speed = total distance ÷ total time.
- Stacked bars compare totals; 100% stacked bars compare proportions.
14. Exam-pattern Questions with Detailed Answers
Find the weighted average of 10 (weight 3) and 20 (weight 2).
Show answer
(30 + 40) ÷ 5 = 14.
Answer: 14
Which chart best compares the percentage of girls in schools of very different sizes?
- Stacked bar
- 100% stacked bar
- Pictograph
- Line graph
Show answer
Proportions are compared best with 100% stacked bars.
Answer: (B)
15 students have mean 40 and 5 students have mean 60. Find the overall mean.
Show answer
(600 + 300) ÷ 20 = 45.
Answer: 45
A test score is computed as 40% theory + 60% practical. Theory = 65, final = 74. Find the practical score.
Show answer
0.4 × 65 + 0.6p = 74 → 26 + 0.6p = 74 → p = 80.
Answer: 80
In what ratio should tea at ₹300/kg be mixed with tea at ₹420/kg to get a mix worth ₹350/kg?
Show answer
(420 − 350) : (350 − 300) = 70 : 50 = 7 : 5.
Answer: 7 : 5
A cyclist rides to a village at 12 km/h and back at 18 km/h. Find the average speed.
Show answer
2 × 12 × 18 ÷ 30 = 14.4 km/h.
Answer: 14.4 km/h
3 litres of 30% juice is mixed with 2 litres of pure water. Find the juice percentage.
Show answer
Juice = 0.9 L in 5 L → 18%.
Answer: 18%
Practice data: Shop A sold 300 items with 20% returned; Shop B sold 100 items with 40% returned. Find the overall return rate.
Show answer
Returns: 60 + 40 = 100 out of 400 → 25%.
Answer: 25%
The mean salary of 40 workers is ₹15,000. When a supervisor joins, the mean rises to ₹15,500. Find the supervisor's salary.
Show answer
New total = 41 × 15500 = 635500. Old total = 600000. Supervisor = ₹35,500.
Answer: ₹35,500
A 100% stacked bar shows School P has 60% girls and School Q has 45% girls. Can we conclude P has more girls? What extra information is needed?
Show answer
No. If P has 200 students (120 girls) and Q has 400 (180 girls), Q has more girls. We need the total number of students in each school.
Answer: No; need the totals
How many litres of water must be added to 12 litres of 25% acid solution to make it 15%?
Show answer
Acid = 3 L. 3 ÷ (12 + w) = 0.15 → 12 + w = 20 → w = 8 L.
Answer: 8 litres
Practice data: Hospital A treats 100 mild cases (90% recover) and 10 serious cases (50% recover). Hospital B treats 20 mild (95% recover) and 80 serious (60% recover). Which hospital has the better overall recovery rate, and which is better for each type? Explain the surprise.
Show answer
A overall: (90 + 5)/110 ≈ 86.4%. B overall: (19 + 48)/100 = 67%. Yet B is better for mild (95% vs 90%) and for serious cases (60% vs 50%). B looks worse overall only because it treats far more serious cases — the weights differ. (This is known as Simpson's paradox.)
Answer: A better overall; B better in each group — due to different weights
15. Quick Revision Summary
- Weighted mean = Σ(w·x) ÷ Σw; combine groups using totals.
- The weighted mean lies nearer the heavier value; equal weights → simple mean.
- Mixtures: weight by quantity; speeds: weight by time.
- Stacked bars show totals; 100% stacked bars show shares only.
- Overall rates can mislead when group sizes differ.