📊 Average of Averages
Finding the Mean of Two Averages — Self-Study Workbook
✏️ Level: Beginner | 10 Word Problems + Commentary
🔑 Super-Easy Memory Keywords — say these out loud!
average= sum ÷ count
meansame as average
combineput together
totaladd everything up
per÷ (divide by)
overallboth groups together
Average =
Sum of all valuesNumber of values
⚠️ TRAP ALERT →
Average of averages ≠ (A + B) ÷ 2 unless group sizes are equal!
📖 Worked Example — Study This First!
✅ Model Example
Group A has 3 students with an average score of 80.
Group B has 5 students with an average score of 70.
What is the overall average score of all students?
Group B has 5 students with an average score of 70.
What is the overall average score of all students?
Step 1 — Find each group's TOTAL (average × count):
Group A total = 80 × 3 = 240
Group B total = 70 × 5 = 350
Step 2 — ADD the totals:
240 + 350 = 590
Step 3 — DIVIDE by total number of people:
3 + 5 = 8 students
Overall average = 5908 = 73.75
❌ Wrong shortcut: (80 + 70) ÷ 2 = 75 ← This ignores group sizes!
Group A total = 80 × 3 = 240
Group B total = 70 × 5 = 350
Step 2 — ADD the totals:
240 + 350 = 590
Step 3 — DIVIDE by total number of people:
3 + 5 = 8 students
Overall average = 5908 = 73.75
❌ Wrong shortcut: (80 + 70) ÷ 2 = 75 ← This ignores group sizes!
✦ ✦ ✦
📝 Practice Problems
→ Write your work below each problem. Try it yourself before reading the commentary!
1
CORE CONCEPT
On Monday, a baker sold 4 cakes at an average price of $12 each.
On Tuesday, she sold 6 cakes at an average price of $8 each.
What was the average price per cake over both days?
MY WORK ✏️
ANSWER: $ ___________
2
⚠️ EQUAL GROUPS EXCEPTION
Team Red has 5 players with an average height of 160 cm.
Team Blue has 5 players with an average height of 170 cm.
What is the average height of all 10 players?
💡 Hint: The group sizes are equal this time. What does that change?
MY WORK ✏️
ANSWER: ___________ cm
📘 Commentary
Red total = 160 × 5 = 800 | Blue total = 170 × 5 = 850Grand total = 800 + 850 = 1650 | Total players = 10
Average = 1650 ÷ 10 = 165 cm
✅ Here (160 + 170) ÷ 2 = 165 also works — only because both groups have the same size (5 = 5)!
3
🔥 TRICKY NUMBERS
In a reading contest, Class A (10 students) read an average of 7 books each.
Class B (4 students) read an average of 3 books each.
How many books did they read on average per student overall?
MY WORK ✏️
ANSWER: ___________ books
📘 Commentary
Class A total = 7 × 10 = 70 | Class B total = 3 × 4 = 12Grand total = 70 + 12 = 82 | Total students = 14
Average = 82 ÷ 14 = ≈ 5.86 books
⚠️ (7 + 3) ÷ 2 = 5 ← Very wrong! Class A has 10 students — it "weighs" much more than Class B with only 4.
4
CORE CONCEPT
A store sold 8 shirts in the morning at an average price of $25,
and 2 shirts in the evening at an average price of $45.
What was the average selling price of a shirt for the whole day?
MY WORK ✏️
ANSWER: $ ___________
📘 Commentary
Morning total = $25 × 8 = $200 | Evening total = $45 × 2 = $90Grand total = $200 + $90 = $290 | Total shirts = 10
Average = 290 ÷ 10 = $29.00
⚠️ (25 + 45) ÷ 2 = $35 ← Way off! 8 morning shirts drag the average much closer to $25.
5
⚠️ COMMON MISTAKE
A farmer harvested apples over two weeks.
Week 1: 3 days of harvesting, average 20 kg per day.
Week 2: 7 days of harvesting, average 30 kg per day.
What was the farmer's average daily harvest over all 10 days?
Week 1: 3 days of harvesting, average 20 kg per day.
Week 2: 7 days of harvesting, average 30 kg per day.
What was the farmer's average daily harvest over all 10 days?
MY WORK ✏️
ANSWER: ___________ kg/day
📘 Commentary
Week 1 total = 20 × 3 = 60 kg | Week 2 total = 30 × 7 = 210 kgGrand total = 60 + 210 = 270 kg | Total days = 10
Average = 270 ÷ 10 = 27 kg/day
⚠️ (20 + 30) ÷ 2 = 25 ← Wrong! Week 2 has 7 days vs only 3. More days = more pull toward 30.
6
🔥 TRICKY NUMBERS
At a pizza party, 6 boys ate an average of 3 slices each,
and 9 girls ate an average of 2 slices each.
What was the average number of slices eaten per person?
MY WORK ✏️
ANSWER: ___________ slices
📘 Commentary
Boys total = 3 × 6 = 18 | Girls total = 2 × 9 = 18Grand total = 18 + 18 = 36 slices | Total people = 15
Average = 36 ÷ 15 = 2.4 slices
⚠️ (3 + 2) ÷ 2 = 2.5 ← Close but wrong! There are 9 girls vs 6 boys, so the answer slides below 2.5.
7
CORE CONCEPT
A runner trains for two weeks. Week 1: she runs 5 days at an average of 6 km/day.
Week 2: she runs 3 days at an average of 10 km/day.
What is her overall average distance per training day?
MY WORK ✏️
ANSWER: ___________ km/day
📘 Commentary
Week 1 total = 6 × 5 = 30 km | Week 2 total = 10 × 3 = 30 kmGrand total = 30 + 30 = 60 km | Total training days = 8
Average = 60 ÷ 8 = 7.5 km/day
⚠️ (6 + 10) ÷ 2 = 8 ← Wrong! 5 shorter days pull the answer below 8.
8
⚠️ COMMON MISTAKE
A school tested two groups in math.
Group 1 (20 students) scored an average of 85 points.
Group 2 (30 students) scored an average of 75 points.
What is the overall average score for all 50 students?
Group 1 (20 students) scored an average of 85 points.
Group 2 (30 students) scored an average of 75 points.
What is the overall average score for all 50 students?
MY WORK ✏️
ANSWER: ___________ points
📘 Commentary
Group 1 total = 85 × 20 = 1700 | Group 2 total = 75 × 30 = 2250Grand total = 1700 + 2250 = 3950 | Total students = 50
Average = 3950 ÷ 50 = 79 points
⚠️ (85 + 75) ÷ 2 = 80 ← Off by 1 point! Group 2 is larger (30 > 20), pulling the answer below 80.
9
🔥 TRICKY NUMBERS
A juice bar serves customers in two shifts.
Morning shift (4 hours): serves an average of 15 customers per hour.
Afternoon shift (6 hours): serves an average of 25 customers per hour.
What is the average number of customers per hour across the full day?
Morning shift (4 hours): serves an average of 15 customers per hour.
Afternoon shift (6 hours): serves an average of 25 customers per hour.
What is the average number of customers per hour across the full day?
MY WORK ✏️
ANSWER: ___________ customers/hour
📘 Commentary
Morning total = 15 × 4 = 60 customers | Afternoon total = 25 × 6 = 150 customersGrand total = 60 + 150 = 210 customers | Total hours = 10
Average = 210 ÷ 10 = 21 customers/hour
⚠️ (15 + 25) ÷ 2 = 20 ← Wrong! The afternoon is 6 hours long — it contributes more to the average.
10
⚠️ SUPER TRAP — looks like a trap, but is it?
A swimmer swims 2 lengths of a pool.
Length 1: takes 2 minutes.
Length 2: takes 3 minutes.
What is the swimmer's average time per length?
💡 Hint: How many lengths are in each group?
MY WORK ✏️
ANSWER: ___________ minutes
📘 Commentary
Length 1 time = 2 × 1 = 2 min | Length 2 time = 3 × 1 = 3 minGrand total time = 2 + 3 = 5 min | Total lengths = 1 + 1 = 2
Average = 5 ÷ 2 = 2.5 minutes per length
✅ Here (2 + 3) ÷ 2 = 2.5 also works — because each group has exactly 1 length (equal!)
🧠 Big Lesson: Always check if the group sizes are equal!
Equal sizes → simple average works ✅
Different sizes → always use (total sum) ÷ (total count) ✅
Equal sizes → simple average works ✅
Different sizes → always use (total sum) ÷ (total count) ✅
✦ ✦ ✦
3 Steps to Never Get It Wrong!
① MULTIPLY average × group size = group TOTAL
② ADD all totals together = GRAND TOTAL
③ DIVIDE by total count = OVERALL AVERAGE
② ADD all totals together = GRAND TOTAL
③ DIVIDE by total count = OVERALL AVERAGE
Overall Average =
(Avg₁ × n₁) + (Avg₂ × n₂)n₁ + n₂
⚡ Only when n₁ = n₂ (equal sizes) can you just do (Avg₁ + Avg₂) ÷ 2
Grand total = $48 + $48 = $96
Total cakes = 4 + 6 = 10
Average = 96 ÷ 10 = $9.60 per cake