✔ 最佳答案
Since I can't type Chinese in this computer, i will try to explain in English.
Actually, your 2 questions are similar to each other. I will look at question 2 first. And I will denote number of 大僧 = B and number of 小僧 = S
Question 2: 點解可以咁計？
Firstly, you have to define "咁計". Here the calculation method involve following steps:
(i) Divide the 100 monks into small group. 4 monks in each group (1 Big monk and 3 small monk)
(ii) Then each group would consume 4 饅頭
(iii) Since 1 group contain 4 monks and consume 4 饅頭, 25 group (=100/4) will contain 100 monks and consume 100 饅頭
(iv) Interestingly, it meets the requirement of number of monks and number of 饅頭 in the question!!!
(v) Therefore, 25 groups, with each group contain 1 big monk and 3 small monk, is the answer of the question. i.e. B=25, S=75
I think what make you puzzled is "when is the question solved?" or "in which step are the question solved?"
I would said the step (i) solve the question. Sub-grouping 1 big monk and 3 small monk together is the key to this answer. This grouping MUST satisfy two conditions for the question to be solved...
(1) There are SAME NUMBER OF monk and 饅頭 in the group
(2) There are 4 monks in a group, which is divisible by 100 (total number of monk)
If anyone of this condition is not met, the question can't be solved.
Let's look closer to these 2 conditions:
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(1) There are SAME NUMBER OF monk and 饅頭 in the group
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This is actually a specify case of the following rules:
(Number of monk in a small group) : (Number of 饅頭 in a small group) = (Total number of monk) : (Total number of 饅頭)
This two ratio must be equal. In this question, the ratio is 1:1
If these two ratio is not equal, we cannot "reproduce" the same number of monk/饅頭 by using small group!
***********************
(2) There are 4 monks in a group. And 4 is divisible by 100 (total number of monk)
***********************
The number of monks in a group must be divisible by total number of monk
If this condition is not met, we cannot "reproduce" the total number of monk by using small group!
OK! Now we can answer to your questions:
Question 1: 係咪要咁岩饅頭同僧數一樣，大小僧分饅頭既比例又要係 x : 1/x 先有得咁計?
No!
Consider the following revised question:
The monk house has grown! There are 160 monks and 200 饅頭. And now every 3 small monk will consume 2 饅頭 (instead of 1), while 1 big monk consume 3 饅頭 as before.
Here, number of monk is not equal to number of 饅頭. And 大小僧分饅頭既比例 is not x : 1/x
We can subgroup the small monk and big monk as followings...
2 Big monk + 6 small monk = 8 monk will consume 10 饅頭. This is one group
Then for 20 groups, there will be 160 monk and 200 饅頭
So there are 20x2 = 40 big monk and 20x6 = 120 small monk
Key is that two conditions are met:
Condition 1: (Number of monk in a small group) : (Number of 饅頭 in a small group) = (Total number of monk) : (Total number of 饅頭) = 4:5
Condition 2: Number of monk in a group (8) is divisible by total number of monk (160)
Although whenever there is a solution, this type of question can be solved in this way, in practise, it is very hard to find the small group needed to solve the question. Using 代數 is much easier... I would said the 100 monk-100 饅頭 quesion is a "very specific example" so that it can be solve in this way. So you may treat it as a “special case”
會唔會有counter example?
I am not sure what do you mean by "counter example". As i said whenever there is a solution, this type of question can be solved in this way. So question with no solution can't be solved in this way!!!
For example, changing the original question to 99 monks and 99 饅頭... no matter how you group, there are no solution!! (Obviously enough, since this question has no solution ><)
