Five friends who play badminton surveyed the city and collected the price of shuttlecocks across brands and types (Nylon and Feather). The following is the list of the prices, in rupees (₹), that each one compiled.
Yusuf: 40(N), 105(N), 383(F), 108(N), 165(F), 116(F)
Srikanth: 194(N), 85(N), 93(N), 121(N)
Kashvi: 49(N), 297(F), 105(N), 275(F), 40(N)
Prasanna: 124(N), 333(F), 182(N), 258(F)
Gracy: 220(F), 458(F), 129(F), 183(N)
Can you describe a way they can work together to find the average price of a shuttlecock across types?
- Opening question Can you describe a way they can work together to find the average price of a shuttlecock across types?
- (i) Each one calculated the average of the prices they had gathered. Suppose these are ay, as, ak, ap, ag respectively. Write an expression that gives the combined average.
- (ii) Suppose an, af are the average prices of the nylon shuttlecocks and the feather shuttlecocks, respectively. Write an expression that gives the average price of a shuttlecock. Will this be equal to the answer we got using the method in the previous part?
Answer: Each friend reports the sum (or average) of their prices and how many they found; they add the sums and divide by the total count, 23. (i) 6ay + 4as + 5ak + 4ap + 4ag23. (ii) 13an + 10af23; yes, both equal 406323 ≈ ₹176.65.
Step-by-step solution
Given: N = nylon shuttle, F = feather shuttle; Counts: Yusuf 6, Srikanth 4, Kashvi 5, Prasanna 4, Gracy 4 prices (23 in all); 13 nylon and 10 feather
Idea: Averages of groups of different sizes must be weighted by the group sizes. However the 23 prices are grouped (by friend or by type), the combined average is always the sum of all 23 prices divided by 23.
Opening question Can you describe a way they can work together to find the average price of a shuttlecock across types?
- Each friend adds up their own prices and counts them (or finds their average and multiplies by their count). They then add the five sums and divide by the total number of prices, 6 + 4 + 5 + 4 + 4 = 23. (They should also remove any shuttlecock brand counted twice by two friends, if it is the same item at the same shop.)1 mark
Add all the friends’ totals and divide by the total number of prices (23).
(i) Each one calculated the average of the prices they had gathered. Suppose these are ay, as, ak, ap, ag respectively. Write an expression that gives the combined average.
- Each friend’s total = average × count: 6ay, 4as, 5ak, 4ap, 4ag.1 mark
- Combined average = 6ay + 4as + 5ak + 4ap + 4ag6 + 4 + 5 + 4 + 4. With the data: ay = 9176, as = 4934, ak = 7665, ap = 8974, ag = 9904, giving 406323 ≈ ₹176.65.1 mark
6ay + 4as + 5ak + 4ap + 4ag23 ≈ ₹176.65
(ii) Suppose an, af are the average prices of the nylon shuttlecocks and the feather shuttlecocks, respectively. Write an expression that gives the average price of a shuttlecock. Will this be equal to the answer we got using the method in the previous part?
- There are 13 nylon prices (sum ₹1429) and 10 feather prices (sum ₹2634). Average price = 13an + 10af13 + 10, with an = 142913 ≈ 109.92 and af = 263410 = 263.4.1 mark
- 13an + 10af = 1429 + 2634 = 4063, so the average is 406323 ≈ ₹176.65. Yes, it is equal: both methods add the same 23 prices and divide by 23; only the grouping is different.1 mark
13an + 10af23 ≈ ₹176.65; yes, the same
Combine the friends’ totals and counts: (i) (6a_y + 4a_s + 5a_k + 4a_p + 4a_g)/23; (ii) (13a_n + 10a_f)/23. Both equal 4063/23 ≈ ₹176.65, because each is the sum of all 23 prices divided by 23.
Check: Sum by friends: 917 + 493 + 766 + 897 + 990 = 4063; sum by type: 1429 + 2634 = 4063 ✓.
Answer to write in the exam
Opening question
Each friend: sum of prices and number of prices
Average = sum of all five totals6 + 4 + 5 + 4 + 4
∴ Combine totals and counts, not just the five averages
(i)
Combined average = 6ay + 4as + 5ak + 4ap + 4ag6 + 4 + 5 + 4 + 4
= 917 + 493 + 766 + 897 + 99023 = 406323
∴ Combined average ≈ ₹176.65
(ii)
Nylon: 13 prices, sum ₹1429; feather: 10 prices, sum ₹2634
Average = 13an + 10af13 + 10 = 1429 + 263423 = 406323
∴ Average ≈ ₹176.65, equal to the answer in (i)
Common mistakes that cost marks
- Writing ay + as + ak + ap + ag5. The friends found different numbers of prices (6, 4, 5, 4, 4).
- Writing an + af2 ≈ 186.66. There are 13 nylon and only 10 feather prices.
- Miscounting the nylon and feather prices; check that 13 + 10 = 23.
How this can come in the exam
MCQ (1 mark)
Three students recorded 5, 3 and 2 prices with averages ₹20, ₹30 and ₹50. The average of all the prices is
- ₹33.33
- ₹29
- ₹30
- ₹100
Show answer
(B) ₹29
5 × 20 + 3 × 30 + 2 × 5010 = 100 + 90 + 10010 = ₹29.
Try one yourself
A shop has 8 pens with an average price of ₹15 and 12 pens with an average price of ₹25. Find the average price of all the pens.
Show answer
8 × 15 + 12 × 2520 = 120 + 30020 = ₹21.
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