Why did Method 1 not work?
Step-by-step solution
Idea: An average stands for the whole group behind it. When groups have different sizes, a bigger group must count more in the combined average.
- Method 1 works out 165.5 + 149.332. Dividing by 2 means it counts the seniors’ average once and the juniors’ average once, as if there were equally many seniors and juniors.½ mark
- In the group, each of the 11 trainees should count once. So the seniors’ average stands for 8 people and should count 8 times; the juniors’ average stands for 3 people and should count 3 times.½ mark
- Counting them correctly: 165.5 × 8 + 149.33 × 38 + 3 = 177211 = 161.09 cm. Method 1’s 157.415 cm is too low, because it gives the 3 shorter juniors too much weight.½ mark
- So the average of two averages is not the combined average when the groups are of different sizes. Each average must be weighted by the size of its group.½ mark
Answer to write in the exam
Method 1 = 165.5 + 149.332 gives equal weight to both groups
But seniors = 8 and juniors = 3, so the groups are of unequal size
Correct average = 165.5 × 8 + 149.33 × 311 = 177211 = 161.09 cm
∴ Method 1 fails because it ignores the different group sizes
Common mistakes that cost marks
- Saying Method 1 failed because of rounding 149.33. The error (about 3.7 cm) is far too big for rounding; it comes from ignoring group sizes.
- Thinking an average of averages is always wrong. It is right when all groups have the same size.
- Explaining only with words like ‘it is wrong’ without showing what the correct weighting (8 and 3) should be.
How this can come in the exam
Assertion (A): If 40 students have an average of 60 and 10 students have an average of 80, the average of all 50 students is 70.
Reason (R): When groups differ in size, each group’s average must be weighted by the size of the group.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(D) A is false but R is true.
Correct average = 40 × 60 + 10 × 8050 = 320050 = 64, not 70, so A is false. R is true.
Try one yourself
Section P has 10 students with an average of 50 marks and Section Q has 30 students with an average of 90 marks. Riya says the combined average is 70. Is she right?
Show answer
No. Combined average = 10 × 50 + 30 × 9040 = 500 + 270040 = 320040 = 80. Riya averaged the two averages, ignoring that Section Q is three times as large.
More questions like this
- 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?
- A bowl of 500 mL lemonade has 10% jaggery. A glass of 200 mL lemonade has 20% jaggery. If we mix the lemonade from the bowl and the glass, what is the concentration of jaggery in the mixture?
Can you estimate what part of the mixture is jaggery? Is it 15%, or is it more or less? Why do you think so?