Average Word Problems
Master challenging problems using the average formulas
The Golden Rule
Always find the TOTAL first, then work from there!
1. Finding Missing Values
Strategy
- Find the total: Total = Average × Number
- Find the missing value: Missing = Total - Sum of known values
Example 1: Finding the Fourth Number
The average of 4 numbers is 20. The sum of 3 of the numbers is 69. What is the fourth number?
Step 1: Find the total
Sum of 4 numbers = Average × Number = 20 × 4 = 80
Step 2: Find the missing number
Fourth number = 80 - 69 = 11
Check: If the 4th number is 11, then sum = 69 + 11 = 80. Average = 80 ÷ 4 = 20 ✓
Example 2: Finding the Fifth Number
The average of 5 numbers is 50. The sum of 4 of the numbers is 192. What is the fifth number?
Step 1: Sum of 5 numbers = 50 × 5 = 250
Step 2: Fifth number = 250 - 192 = 58
2. Problems with Bar Graphs
Strategy
- Find total: Total = Average × Number of days (or items) the average is for. Count the missing one too, even if it has no bar.
- Add visible bars: Sum the bars you can read
- Find missing: Missing bar = Total - Sum of visible bars
Example: Packet Drinks Sold
The bar graph shows the number of packet drinks sold by a drink seller over 4 days. The average number of packets sold each day was 48. The bar showing Thursday is torn off. How many packets were sold on Thursday?
Given: Mon = 56, Tue = 44, Wed = 52, Thu = ?
Average = 48 packets per day over 4 days
Step 1: Find total packets sold
Total = 48 × 4 = 192 packets
Step 2: Sum of visible bars
Mon + Tue + Wed = 56 + 44 + 52 = 152 packets
Step 3: Find Thursday
Thursday = 192 - 152 = 40 packets
Example: Patients at Clinic
The bar graph shows the number of patients at Lim's Clinic each day from Monday to Friday. The average number of patients a day from Monday to Saturday was 38. How many patients were there on Saturday?
Step 1: Total for 6 days (Mon-Sat) = 38 × 6 = 228
Step 2: Sum Mon-Fri = 44 + 58 + 32 + 52 + 26 = 212
Step 3: Saturday = 228 - 212 = 16 patients
3. Combined Averages
Important Warning!
You CANNOT simply add averages!
If boys average 7 and girls average 12, the combined average is NOT 7 + 12 = 19! It is not (7 + 12) ÷ 2 = 9.5 either, unless there are as many boys as girls. Always go back to the totals.
Strategy
- Find combined total = Combined average × Total count
- Find the total of the group you know = that group's average × number in that group
- Find the other group's total = Combined total - Known group's total
Example: Boys and Girls Savings
In March, 3 boys and 2 girls saved an average of $10 each. The 3 boys saved an average of $7 each. How much did the 2 girls save in total?
Step 1: Total saved by all 5 children
= $10 × 5 = $50
Step 2: Total saved by 3 boys
= $7 × 3 = $21
Step 3: Total saved by 2 girls
= $50 - $21 = $29
Example: Buying Fruits
4 men and 6 women bought an average of 12 fruits each. The 4 men bought an average of 10 fruits each. How many fruits did the 6 women buy in total?
Step 1: Total fruits bought by all 10 people
= 12 × 10 = 120 fruits
Step 2: Total fruits bought by 4 men
= 10 × 4 = 40 fruits
Step 3: Total fruits bought by 6 women
= 120 - 40 = 80 fruits
4. New Member Joins
Strategy
- Find old total = Old average × Old count
- Find new total = Old total + New member's value
- Find new average = New total ÷ New count
Example: Average Height Changes
The average height of 4 children is 1.45 m. Another child whose height is 1.55 m joins the group. What is the average height of the 5 children?
Step 1: Total height of 4 children
= 1.45 m × 4 = 5.8 m
Step 2: Total height of 5 children
= 5.8 m + 1.55 m = 7.35 m
Step 3: New average height
= 7.35 m ÷ 5 = 1.47 m
Notice: The new child (1.55 m) is taller than the old average (1.45 m), so the new average (1.47 m) goes UP!
Example: Adding a Ribbon
Hasnah has 3 ribbons. The average length of the 3 ribbons is 8.5 m. She buys another ribbon that is 7.5 m long. What is the average length of all the ribbons?
Step 1: Total length of 3 ribbons = 8.5 m × 3 = 25.5 m
Step 2: Total length of 4 ribbons = 25.5 m + 7.5 m = 33 m
Step 3: New average = 33 m ÷ 4 = 8.25 m
Notice: The new ribbon (7.5 m) is shorter than the old average (8.5 m), so the new average (8.25 m) goes DOWN!
5. Challenging Problems
Example: Teacher Joins the Class
The average mass of 39 students in Class 6A is 40 kg. When the mass of the teacher is included, the average mass becomes 41 kg. What is the mass of the teacher?
Step 1: Total mass of 39 students
= 40 × 39 = 1560 kg
Step 2: Total mass of 40 people (39 students + teacher)
= 41 × 40 = 1640 kg
Step 3: Mass of teacher
= 1640 - 1560 = 80 kg
Key Problem-Solving Strategies
- 1.Missing Values: Total = Average × Number, then Missing = Total - Known sum
- 2.Bar Graphs: Total = Average × Number of items the average is for, then Missing bar = Total - Sum of the bars shown
- 3.Combined Groups: Find both totals separately, then subtract
- 4.New Member: New total = Old total + New value, then divide by new count
- !Golden Rule: Always find the TOTAL first!