Measures of central tendency · Ages 14–15
Weighted mean word problems
Weight a combined mean by the group sizes.
How to do it
Method
Weight a combined mean by the group sizes
(20 × 60 + 30 × 70) ÷ 50 = 66
Another way
The gap in proportion
60 + (70 − 60) × 3050 = 66
In full
To combine two groups' means, never just average the two means: weight each by its size. Team A has 20 people averaging 60 kg and team B 30 averaging 70 kg; together, (20 × 60 + 30 × 70) ÷ (20 + 30) = 3300 ÷ 50 = 66 kg, not 65 kg. A second way: start from the smaller mean and add the gap in proportion to the other group's share: 60 + (70 − 60) × 3050 = 66. Working back, from a combined mean to one group's mean or size: find the totals and subtract, or set up an equation. For a combined mean of 66, how many in team B? 20 × 60 + 70y = 66(20 + y) gives y = 30.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Fleet A's 18 vans have a mean age of 39 months and fleet B's 49 months. Together their mean age is 44 months. How many vans are in fleet B?
Show the answer and working
Answer: 18 vans
Say fleet B has y: 18 × 39 + 49y = 44(18 + y), so 18 × (44 − 39) = y × (49 − 44): y = 18.
Example 2
A running club's 24 members ran 52 km each on average last week. Its 12 men averaged 30 km. How many km did its 12 women average?
Show the answer and working
Answer: 74 km
In all 24 × 52 = 1248 km, the men 12 × 30 = 360 km, so the women (1248 − 360) ÷ 12 = 74 km.
Example 3
13 morning customers spent $55 each on average and 19 afternoon customers $63. What did a customer spend on average all day?
- A. $59
- B. $58.25
- C. $59.75
- D. $956
Choose the right answer
Show the answer and working
Answer: C. $59.75
(13 × 55 + 19 × 63) ÷ (13 + 19) = 59.75: weighted by the numbers of customers.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Group A's 5 people folded 11 paper cranes each on average, and group B's 6 people 7 each. How many cranes did the two groups fold in all?
Answer:
2.Team A's 11 people have a mean weight of 41 kg and team B's 39 people 48 kg. What is the mean weight of the two teams together, in kg?
Answer:
3.Fleet A's 18 vans have a mean age of 41 months and fleet B's 60 months. Together their mean age is 54 months. How many vans are in fleet B?
Answer:
4.A running club's 36 members ran 49 km each on average last week. Its 24 men averaged 52 km. How many km did its 12 women average?
Answer:
5.Group A's 13 people take 46 minutes on average to finish a jigsaw and group B's 13 people 66 minutes. The two groups' combined mean time is exactly the mean of 46 and 66.
True or false?
Answer:
All the answers
- 1. 97 cranes
- 2. 46.46 kg
- 3. 39 vans
- 4. 43 km
- 5. True
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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