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primary 6 | Maths
| Data Analysis
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Hi, please help on 14(b). Thanks.
Date Posted: 2 years ago
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This is actually a pattern spotting question.
14a) There is actually a simpler and faster way rather than to add up all eight visible numbers, and then divide by 8, if you spot the pattern.
What we realise is that the middle number 21 (covered by the grey square of the frame) is the average of all 9 numbers in the frame.
Notice that for the other eight numbers in the frame surrounding 21 that can be seen, we can get these four pairs:
① 30 = 21 + 9 and 12 = 21 - 9 (left diagonal numbers)
② 29 = 21 + 8 and 13 = 21 - 8 (middle vertical)
③ 28 = 21 + 7 and 14 = 21 - 7 (right diagonal)
④20 = 21 - 1 and 22 = 21 + 1 (middle horizontal)
In each pair, the additions and subtractions will cancel out, so we will get two times of 21 when we add the two numbers.
Eg. 30 + 12 = 21 + 9 + 21 - 9 = 21 + 21 = 2 × 21 = 42
All 4 pairs add up to eight 21's, so the average of the eight numbers is just 21.
And since the middle number is 21 also, then the average of all nine numbers in the frame is also 21. We don't actually have to calculate anything.
14b) Firstly, find the average of the eight numbers. 424 ÷ 8 = 53
So the middle number is 53 also (see above explanation if you're unsure why it is also 53)
Next, we can see from 14a) that the biggest number is 8 larger than the middle.
The smallest number is 8 smaller than the middle.
So their sum = 53 + 8 + 53 - 8 = 53 + 53 = 53 × 2 = 106 (if you understand this, you can just write 53 × 2 for the working)
We saw from 14a) that the sum of each pair is 2 times the average of all eight (or all nine numbers . This sum of the largest and smallest is similar to the pair ① you see above (also sum of largest and smallest in the frame)
14a) There is actually a simpler and faster way rather than to add up all eight visible numbers, and then divide by 8, if you spot the pattern.
What we realise is that the middle number 21 (covered by the grey square of the frame) is the average of all 9 numbers in the frame.
Notice that for the other eight numbers in the frame surrounding 21 that can be seen, we can get these four pairs:
① 30 = 21 + 9 and 12 = 21 - 9 (left diagonal numbers)
② 29 = 21 + 8 and 13 = 21 - 8 (middle vertical)
③ 28 = 21 + 7 and 14 = 21 - 7 (right diagonal)
④20 = 21 - 1 and 22 = 21 + 1 (middle horizontal)
In each pair, the additions and subtractions will cancel out, so we will get two times of 21 when we add the two numbers.
Eg. 30 + 12 = 21 + 9 + 21 - 9 = 21 + 21 = 2 × 21 = 42
All 4 pairs add up to eight 21's, so the average of the eight numbers is just 21.
And since the middle number is 21 also, then the average of all nine numbers in the frame is also 21. We don't actually have to calculate anything.
14b) Firstly, find the average of the eight numbers. 424 ÷ 8 = 53
So the middle number is 53 also (see above explanation if you're unsure why it is also 53)
Next, we can see from 14a) that the biggest number is 8 larger than the middle.
The smallest number is 8 smaller than the middle.
So their sum = 53 + 8 + 53 - 8 = 53 + 53 = 53 × 2 = 106 (if you understand this, you can just write 53 × 2 for the working)
We saw from 14a) that the sum of each pair is 2 times the average of all eight (or all nine numbers . This sum of the largest and smallest is similar to the pair ① you see above (also sum of largest and smallest in the frame)
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