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?
Step-by-step solution
Idea: An average can be ‘undone’: multiplying an average by the number of values gives back the sum of those values. Then the combined average is the total sum divided by the total number.
- Average of the seniors = sum of seniors’ heights8 = 165.5. Multiply both sides by 8: sum of seniors’ heights = 165.5 × 8 = 1324 cm. (Check: 165 + 169 + 164 + 167 + 170 + 159 + 164 + 166 = 1324 ✓.)½ mark
- In the same way, sum of the juniors’ heights = 149.33 × 3 = 447.99 ≈ 448 cm (the exact sum is 146 + 149 + 153 = 448; 149.33 is a rounded average).½ mark
- So the numerator 1324 + 448 = 1772 is the sum of all 11 heights, and the denominator 8 + 3 = 11 is the number of trainees.½ mark
- Sum of all heights ÷ number of trainees is exactly the definition of the average, so Shreyas’s method gives the correct 1772 ÷ 11 = 161.09 cm. In general, two groups with averages a, b and sizes n, m have combined average an + bmn + m.½ mark
Answer to write in the exam
Average = sum of valuesnumber of values ⇒ sum = average × number
Sum of seniors’ heights = 165.5 × 8 = 1324 cm
Sum of juniors’ heights = 149.33 × 3 ≈ 448 cm
Average of group = 1324 + 4488 + 3 = 177211 = 161.09 cm
∴ Shreyas’s method gives the total of all heights ÷ total number, so it is correct
Common mistakes that cost marks
- Thinking 149.33 × 3 = 447.99 means the method is slightly wrong. 149.33 is rounded; the exact juniors’ total is 448.
- Dividing by 2 at the end instead of by 8 + 3 = 11.
- Multiplying each average by the other group’s size (165.5 × 3), which gives a meaningless number.
How this can come in the exam
The average of 12 numbers is 25. The sum of the 12 numbers is
- 37
- 300
- 250
- 2.08
Show answer
(B) 300
Sum = average × number = 25 × 12 = 300.
Try one yourself
The average weight of 5 bags is 18 kg and of 7 other bags is 24 kg. Find the total weight of all 12 bags and their average weight.
Show answer
Totals: 5 × 18 = 90 kg and 7 × 24 = 168 kg, so all 12 bags weigh 258 kg. Average = 258 ÷ 12 = 21.5 kg.
More questions like this
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