📑 On this page
⚡ The short answer
Weighted average = (value₁ × weight₁ + value₂ × weight₂ + …) ÷ (weight₁ + weight₂ + …). Use it whenever you combine groups of different sizes. The trap: averaging the averages, which works only for equal groups. The answer always sits between the group values, nearer the bigger group.
🎬 The 30-second story
The 4.3-star banner
1
“Outlet A: 4.8 stars. Outlet B: 3.8 stars. Brand average?”
🍜
Kabir
Kabir runs two cloud-kitchen outlets and wants a banner.
2
“Easy. (4.8 + 3.8) ÷ 2 = 4.3. Print it!”
🧮
Aarav
Aarav does it in three seconds.
3
“A has 40 reviews, B has 160. They don't count equally.”
📱
Zoya
Zoya checks how many people stand behind each rating.
4
“So my banner was lying by 0.3 of a star?”
🪧
Kabir
The printer is told to wait.
Tara's maths: Total stars = 40 × 4.8 + 160 × 3.8 = 192 + 608 = 800 from 200 reviews, so the honest rating is 800 ÷ 200 = 4.0. Outlet B has four times the reviews, so it gets four times the say.
What is a weighted average, and why can't you just average the averages?
A plain average assumes every value speaks with one voice. But Kabir's 4.8 is not one review; it summarises 40 reviews, and his 3.8 summarises 160. Averaging 4.8 and 3.8 secretly pretends both outlets have the same number of reviewers. A weighted average lets each value count as many times as the people, tonnes or hours it stands for.
🟦 Picture the percent
200 reviews, shrunk to 100 squares
4.0★ overall
Only 1 square in 5 carries the high rating.
4.8★ outlet (40 reviews)3.8★ outlet (160 reviews)
Averaging 4.8 and 3.8 would give each colour 50 squares. The real split is 20 : 80.
The whole idea in one line: rebuild the totals, then divide. Every average hides a total (average × count). Add the hidden totals, divide by the total count, and the weights look after themselves.
What is the weighted average formula?
🧮 The formula
Weighted average, built from totals
🧺
Σ (value × weight)
each group's hidden total
÷
⚖️
Σ weights
how many in all
=
🎯
Weighted average
the honest overall
For two groups: (w₁ × x₁ + w₂ × x₂) ÷ (w₁ + w₂).
The weight is how much of each thing there is: students, tonnes, hours, orders. The value is the per-unit figure: marks per student, ₹ per tonne, km per hour. Value × weight must give a sensible total. If it doesn't, you picked the wrong weight.
📌 Weights only need to be in the right proportion
40 and 160 reviews can be written as 1 : 4 or 20% : 80%, and the answer is the same: (1 × 4.8 + 4 × 3.8) ÷ 5 = 20 ÷ 5 = 4.0. So when CLAT gives percentage shares or a ratio instead of counts, use them directly as weights.
🎛️ Try it yourself — change any number
Kabir's outlets: play with the weights
Weighted average
4
Simple average (wrong here)
4.3
(40×4.8 + 160×3.8) ÷ 200 = 4
Set both review counts to 100 and the two averages match. Push outlet B to 1,000 reviews and watch the overall crawl towards 3.8.
How do you calculate a weighted average step by step?
✍️ Worked example
Warm-up
⏱ 40 sec
Meher's semester grade
Meher scores grade point 9 in Maths (4 credits), 8 in English (3 credits) and 10 in Sports (1 credit). What is her semester grade point average?
- 1The weights are the credits: 4, 3 and 1. Total weight = 4 + 3 + 1 = 8.
- 2Value × weight for each subject: 9 × 4 = 36, 8 × 3 = 24, 10 × 1 = 10.
- 3Sum of products = 36 + 24 + 10 = 70.
- 4Weighted average = 70 ÷ 8 = 8.75.
Answer: 8.75, not the simple average of 9. Sports carries 1 credit, so its 10 barely moves the result.
⚡ The exam-hall shortcut
It must lie between 8 and 10 and lean towards Maths and English, which hold 7 of the 8 credits.
✍️ Worked example
CLAT level
⏱ 60 sec
The fest's average ticket price
A college fest sells 300 early-bird tickets at ₹150, 500 regular tickets at ₹250 and 200 gate tickets at ₹400. What is the average price paid per ticket?
- 1Total tickets (the weight) = 300 + 500 + 200 = 1,000.
- 2Revenue: 300 × 150 = ₹45,000; 500 × 250 = ₹1,25,000; 200 × 400 = ₹80,000. Total ₹2,50,000.
- 3Average price = 2,50,000 ÷ 1,000 = ₹250.
Answer: ₹250, not the ₹266.67 you get by averaging 150, 250 and 400.
⚡ The deviation shortcut
Measure from ₹250. Early-bird: −100 × 300 = −30,000. Regular: 0. Gate: +150 × 200 = +30,000. They cancel, so the average is ₹250 with no division.
✍️ Worked example
Tricky
⏱ 75 sec
The missing premium price
60% of the phones a brand sold were budget models at an average of ₹12,000. The rest were premium. The average price of all phones sold was ₹21,000. What was the average price of a premium phone?
- 1Weights are percentages: 60 budget, 40 premium per 100 phones.
- 2Total for 100 phones = 100 × 21,000 = ₹21,00,000. Budget phones bring 60 × 12,000 = ₹7,20,000.
- 3Premium phones bring 21,00,000 − 7,20,000 = ₹13,80,000 over 40 phones = ₹34,500 each.
Answer: ₹34,500.
⚡ The see-saw shortcut
Distances are the inverse of weights. Weights 3 : 2, so distances 2 : 3. Budget sits 9,000 below, which is 2 parts, so 1 part = 4,500. Premium sits 3 parts = 13,500 above: ₹34,500.
Is there a faster way? The see-saw (deviation) method
Picture a see-saw. A heavy kid sits close to the pivot, a light kid far out, and it balances. The weighted average is the pivot, and the bigger group is the heavy kid. The balance rule: weight × distance is equal on both sides. Kabir: 40 × (4.8 − 4.0) = 32 and 160 × (4.0 − 3.8) = 32.
✖️ The alligation cross
Kabir's outlets on the cross
Outlet B : Outlet A = 4 : 1
Each outlet's share is the other outlet's distance from the overall: 0.8 : 0.2 = 4 : 1, exactly 160 : 40 reviews. That is alligation: a weighted average solved backwards.
💡 Estimate first, then let the options do the work
Before calculating, ask: between which values must it lie? and which side is heavier? With weights 1 : 4, the answer sits one-fifth of the way from the heavy value. That often leaves one option standing.
How does CLAT hide a weighted average inside a caselet?
CLAT never says "find the weighted average". It gives a paragraph: a chain sells in three cities, each with its own orders and average bill, then asks for the bill across the chain. Watch for overall, taken together or combined. Those words mean: rebuild totals.
📊 The data
A cloud-kitchen chain's average bill per order
Delhi · 1,200 orders
₹320
Pune · 600 orders
₹410
Jaipur · 200 orders
₹260
Simple average (wrong)
₹330
Whole chain (weighted) · 2,000 orders
₹341
Pune's high bill and big weight drag the true figure above the simple average.
✍️ Worked example
Caselet
⏱ 50 sec
The chain's average bill
Using the figures in the chart, what is the average bill per order across the whole chain?
- 1Revenue by city: Delhi 1,200 × 320 = ₹3,84,000; Pune 600 × 410 = ₹2,46,000; Jaipur 200 × 260 = ₹52,000.
- 2Total revenue = ₹6,82,000. Total orders = 1,200 + 600 + 200 = 2,000.
- 3Average bill = 6,82,000 ÷ 2,000 = ₹341.
Answer: ₹341 per order, not the ₹330 trap.
⚡ The exam-hall shortcut
Base ₹320 (Delhi). Pune: +90 × 600 = +54,000. Jaipur: −60 × 200 = −12,000. Net +42,000 ÷ 2,000 = +21, so ₹341.
| Group | Students | Pass % | Passed |
|---|---|---|---|
| Boys | 200 | 60% | 120 |
| Girls | 300 | 80% | 240 |
| Whole school | 500 | 72% | 360 |
→ scroll to see more
Percentages are averages too. An overall pass rate is a weighted average of group pass rates, weighted by group size: 60% and 80% make 72% here, not 70%. For any overall percentage, go back to counts: total "yes" ÷ total base.
What traps does CLAT set with weighted averages?
⚔️ Side by side
Simple average vs weighted average
🟰Simple average
Every value counts once
- •Right for a raw list of values
- •Right for equal groups
🏋️Weighted average
Each value counts as often as it stands for
- •Needed when group sizes differ
- •Weights can be counts, ratios or % shares
VS
| Test | Simple average | Weighted average |
|---|---|---|
| Formula | Sum ÷ count | Σ(value × weight) ÷ Σ weights |
| Kabir's outlets | 4.3★ (wrong) | 4.0★ |
With equal groups both give the same answer. That is the only time the shortcut is safe.
🧭 Decision flow
Which average does this question want?
❓ Are you combining averages, rates or percentages of groups?
No →
Plain average: sum ÷ count.
Yes ↓
❓ Are the groups the same size?
Yes →
Averaging the averages is safe here.
No ↓
❓ Do you know the sizes, or at least their ratio or % share?
Yes →
Weighted average: rebuild the totals, then divide.
No →
Cannot be determined. You only know a range.
CLAT's favourite option D.
⚠️ Weight by the thing in the denominator
Price per tonne is weighted by tonnes; speed in km per hour by hours. Drive 2 hours at 40 km/h and 1 hour at 70 km/h: (2 × 40 + 1 × 70) ÷ 3 = 50 km/h, not 55. Whatever follows "per" is your weight.
How did CLAT ask weighted average questions?
📝 Asked in CLAT 2022
· Q145
Overall average = total ÷ total
An Indian company… produces five products namely P, Q, R, S and T. Total production of the company for the financial year 2021-22 is 3,000 tonnes and the turnover of the company is ₹50 million. … Production of product 'P' is 21% of the total production and 18% of the turnover is attributable to product 'P'; production of 'Q' is 16% of the total production and 17% of the turnover is attributable to 'Q'; 'R' accounts for 18% of the total production and 20% of the turnover; 'S' accounts for 20% of the total production and 25% of the turnover, and; 'T' accounts for 25% of total production and 20% of turnover.
What is the average selling price per tonne of all products taken together?
A₹17,488
B₹17,667
C₹18,667
D₹16,667
Reveal answer + reasoning
Answer: D
"Taken together" means rebuild totals: ₹5,00,00,000 ÷ 3,000 tonnes = ₹16,666.67 ≈ ₹16,667. The long route agrees. Per-tonne prices are P ≈ ₹14,286, Q ≈ ₹17,708, R ≈ ₹18,519, S ≈ ₹20,833, T ≈ ₹13,333; weighted by tonnes (21, 16, 18, 20, 25%) they give ₹16,667. A plain average of the five gives about ₹16,936, the trap a tougher paper would offer.
Time: 20–30 seconds.
Why the others fail
- A — Above ₹16,667; total ÷ total cannot give it.
- B — ₹1,000 too high: a slip in 5,00,00,000 ÷ 3,000.
- C — ₹2,000 too high; even the plain average of the five prices (about ₹16,936) doesn't reach it.
📝 Asked in CLAT 2020
· Q139
Comparing two averages
The following data presents the cases of Japanese Encephalitis in five states of India. Number of persons affected (in thousands): Odisha 57, West Bengal 45, Chhattisgarh 21, Jharkhand 29, Andhra Pradesh 13. Ratio of males to females: Odisha 12 : 7, West Bengal 6 : 3, Chhattisgarh 5 : 2, Jharkhand 15 : 14, Andhra Pradesh 7 : 6.
By what percentage is the average of males in Chhattisgarh, Jharkhand and Andhra Pradesh together more or less than the average of females in West Bengal, Jharkhand and Odisha together who are affected by Japanese encephalitis?
A22
B24
C26
D28
Reveal answer + reasoning
Answer: C
Split each state by its ratio (thousands). Males: Chhattisgarh 21 × 5/7 = 15, Jharkhand 29 × 15/29 = 15, Andhra Pradesh 13 × 7/13 = 7; total 37. Females: West Bengal 45 × 3/9 = 15, Jharkhand 29 × 14/29 = 14, Odisha 57 × 7/19 = 21; total 50. Both averages are over 3 states, so the ÷3 cancels: (50 − 37) ÷ 50 × 100 = 26% less.
Time: about 90 seconds.
Why the others fail
- A — 22% would need a gap of 11 thousand; the gap is 13.
- B — 24% would need a gap of 12 thousand.
- D — 28% would need 14 thousand. The wrong base (37) gives about 35%, not offered.
Check yourself: myths and a five-question quiz
💥 Myth vs fact
Weighted average myths
🙅 Myth
The overall average of two groups is the average of their averages.
Tap to bust it →
✅ Fact
Only when the groups are the same size. Otherwise rebuild totals.
🙅 Myth
Weights must be actual head-counts.
Tap to bust it →
✅ Fact
Ratios and percentage shares work just as well.
🙅 Myth
A weighted average can come out higher than every group's value.
Tap to bust it →
✅ Fact
Never. It always lies between the lowest and highest group values, nearer the heavier group.
🙅 Myth
Alligation is a separate topic to memorise.
Tap to bust it →
✅ Fact
It is the same equation solved for the ratio.
🎯 Quick quiz
Five quick ones
0 / 5
Section A has 25 students averaging 80. Section B has 75 students averaging 60.
Q1. What is the average of all 100 students?
(2,000 + 4,500) ÷ 100 = 65. 70 is the averaging-the-averages trap.A juice stall sells glasses at ₹40 and ₹60 in the ratio 3 : 2.
Q2. What is the average price per glass sold?
(120 + 120) ÷ 5 = ₹48.A class of 50 averages 72. Its 20 boys average 66.
Q3. What do the 30 girls average?
3,600 − 1,320 = 2,280; ÷ 30 = 76.Group X averages 30 and Group Y averages 50. Group Y has more members than Group X.
Q4. Which of these could be the combined average?
Equal groups give 40. Y is bigger, so it must be above 40: only 44.An office has 400 staff in Mumbai, 25% of them women, and 600 staff in Pune, 40% of them women.
Q5. What percentage of all staff are women?
(100 + 240) ÷ 1,000 = 34%. 32.5% is the simple average trap.🎯 Weighted average in a nutshell
- Total = average × count. Rebuild totals, then divide.
- Weighted average = Σ(value × weight) ÷ Σ weights.
- It lies between the group values, nearer the bigger group.
- See-saw: weight × distance balances. Backwards, that is alligation.
- No group sizes? Cannot be determined.
📎 Formula card
- Weighted average = (w₁x₁ + w₂x₂ + …) ÷ (w₁ + w₂ + …)
- See-saw: w₁ × (A − x₁) = w₂ × (x₂ − A)
- Ratio form: w₁ : w₂ = (x₂ − A) : (A − x₁)
- Deviation method: A = base + Σ(w × deviation) ÷ Σw
- Overall % = total "yes" ÷ total base
- Average speed = speeds weighted by time
- Check: smallest < A < largest
Frequently asked questions
What is the weighted average formula?
Weighted average = (value₁ × weight₁ + value₂ × weight₂ + …) ÷ (weight₁ + weight₂ + …). Multiply each value by how much of it there is, add the products, and divide by the total weight. For 40 reviews at 4.8 and 160 at 3.8, it is 800 ÷ 200 = 4.0.
What is the difference between a simple average and a weighted average?
A simple average counts every value once. A weighted average counts each value as often as it stands for, such as students, tonnes or hours. They match only when every group is the same size. With unequal groups, the weighted average leans towards the bigger group.
How do I know a CLAT question needs a weighted average?
Look for overall, combined or taken together, applied to groups of different sizes. If the passage gives an average, rate or percentage for each group and asks for one figure across all of them, rebuild the totals and divide.
Can the weights be percentages or ratios?
Yes. Only the relative size of the weights matters, so 40 and 160, 1 : 4 and 20% : 80% all give the same answer. CLAT caselets often give percentage shares, and you can use them directly as weights.
How is weighted average related to alligation?
They are the same equation. A weighted average uses the weights to find the overall value; alligation starts from the overall value and finds the ratio of the weights, as (dearer − mean) : (mean − cheaper). Both come from the see-saw rule.
Keep going
← Previous topic
When Someone Joins or Leaves: How the Average Shifts
Someone joins or leaves: work with totals and find the mover fast
Next topic →
Alligation: the Cross Method
Two prices, one target: the cross gives the mixing ratio in seconds
📚
Part of the chapter
Averages, Mixtures & Alligation for CLAT Quantitative Techniques
See how this topic fits the whole of Averages, Mixtures & Alligation →
Connected topics in other chapters
Alligation: the Cross Method
The same see-saw run backwards: you know the overall, you want the mixing ratio.
Average Speed: Why It Is Not the Average of Speeds
Speeds weighted by time, the most famous weighted-average trap of all.
Average: Formula, Meaning and the Balance Idea
The sum = average × count habit that every weighted average is built on.
Caselet DI: Turning a CLAT Quant Passage into a Table
Where weighted averages actually hide on the exam: inside a paragraph of numbers.
Mixtures, Concentration and Replacement
A mixture's strength is just a weighted average of its ingredients' strengths.
Every topic in Averages, Mixtures & Alligation
- 1.Average: Formula, Meaning and the Balance Idea
- 2.When Someone Joins or Leaves: How the Average Shifts
- 3.Weighted Average: Why You Can't Just Average the Averages (you're here)
- 4.Alligation: the Cross Method
- 5.Mixtures, Concentration and Replacement
- 6.Average Speed: Why It Is Not the Average of Speeds