In a badminton academy, there is a group of 11 trainees — 8 seniors and 3 juniors. Their heights and average heights (in cm) are given in the table below. Find the average height of the whole group.
Two students calculate the average height of the whole group in two ways:
Method 1: 165.5 + 149.332 = 314.832 = 157.415
Method 2: 165 + 169 + 164 + 167 + 170 + 159 + 164 + 166 + 146 + 149 + 15311 = 177211 = 161.09
Whose calculation gives the correct average height of the whole group?
Step-by-step solution
| Heights (in cm) | Average Height (in cm) | |
|---|---|---|
| Seniors | 165, 169, 164, 167, 170, 159, 164, 166 | 165.5 |
| Juniors | 146, 149, 153 | 149.33 |
To find: The average height of all 11 trainees, and which method is correct
Idea: The average of a group is (sum of all the values) ÷ (number of values). Every trainee must count once. Method 1 averages the two group averages, which counts the 3 juniors as heavily as the 8 seniors.
- Average height of the whole group = sum of the heights of all 11 trainees11.½ mark
- Sum of the seniors’ heights = 165 + 169 + 164 + 167 + 170 + 159 + 164 + 166 = 1324 cm.
Sum of the juniors’ heights = 146 + 149 + 153 = 448 cm.1 mark - Sum of all 11 heights = 1324 + 448 = 1772 cm, so the average = 177211 = 161.09 cm.1 mark
- This is exactly Method 2, so Method 2 is correct. Method 1 gives 157.415 cm, which is too low: it gives the 3 short juniors the same say as the 8 taller seniors.½ mark
Check: Seniors: 1324 ÷ 8 = 165.5 ✓ and juniors: 448 ÷ 3 = 149.33 ✓, matching the table. The answer 161.09 lies between 149.33 and 165.5, and closer to 165.5 because there are more seniors ✓.
Answer to write in the exam
Sum of heights of seniors = 165 + 169 + 164 + 167 + 170 + 159 + 164 + 166 = 1324 cm
Sum of heights of juniors = 146 + 149 + 153 = 448 cm
Average height = 1324 + 4488 + 3 = 177211 = 161.09 cm
∴ Method 2 is correct; average height of the group = 161.09 cm
Common mistakes that cost marks
- Adding the two averages and dividing by 2 (Method 1). That treats 3 juniors as if they were as many as 8 seniors.
- Dividing the total 1772 by 2 (the number of groups) or by 8 (the number of seniors) instead of by 11 trainees.
- Copying a height wrongly while adding 11 numbers. Check each group’s sum against its average: 1324 ÷ 8 should give 165.5.
How this can come in the exam
In a class, 20 boys have an average mark of 60 and 30 girls have an average mark of 70. The average mark of the whole class is
- 65
- 66
- 64
- 67
Show answer
(B) 66
20 × 60 + 30 × 7050 = 1200 + 210050 = 330050 = 66.
The average age of 4 teachers on a trip is 40 years and the average age of the 36 students with them is 15 years. Find the average age of all 40 people.
Show answer
Sum of teachers’ ages = 4 × 40 = 160 years; sum of students’ ages = 36 × 15 = 540 years (1 mark).Average = 160 + 54040 = 70040 = 17.5 years (1 mark).
Try one yourself
6 boys have an average height of 150 cm and 4 girls have an average height of 145 cm. Find the average height of all 10 children.
Show answer
6 × 150 + 4 × 14510 = 900 + 58010 = 148 cm.
More questions like this
- Why did Method 1 not work?
- Shreyas calculated it differently as (165.5 × 8) + (149.33 × 3)8 + 3 = 1324 + 44811 = 177211 = 161.09. Do you understand why this also works? Can you see why 165.5 × 8 gives the sum of the heights of all the seniors and 149.33 × 3 gives the sum of the heights of all the juniors?
- Jaspreet recently learnt cycling. She has explored different routes in her town. She tracked how much time she cycled on weekdays over the last 3 weeks. Find the mean time spent cycling per weekday over the last 3 weeks.
- We saw earlier how Method 1 (badminton example) did not produce the correct value. Why does Method 1 give the correct answer in this case? When does Method 1 work and when does it not work?
- Two glasses of equal quantities of lemonade are prepared. One glass has 10% jaggery and the other has 20% jaggery. If we mix the lemonade from both glasses, what is the concentration of jaggery in the mixture?