General solution: (abundance+deficit) divided by two distributions, only the difference of each copy = the number of divided objects, and the number of articles can be calculated by the number of shares and profits and losses of one of the methods.
Other (advanced): the critical point of profit and loss-the base point of stock trading volume of the exchange, beyond which profit will be realized, and vice versa.
Basic model for calculating the critical point of profit and loss
Let P represent profit, V represents sales volume, SP represents unit price, VC represents unit variable cost, FC represents fixed cost, and BE represents break-even point. According to the profit calculation formula, the basic model of breakeven point can be obtained as follows:
The calculation of the break-even point can take two forms:
1. Calculated in physical units:
Among them, the unit price of a product is 10 yuan, the unit variable cost is 6 yuan, and the related fixed cost is 8000 yuan, so the sales volume of the breakeven point (physical unit) = 8000÷( 10-6)= 2000 (pieces). Gross profit of product contribution = unit product sales revenue-unit variable cost
2. Comprehensive calculation by amount: the sales volume of breakeven point (expressed by amount) = fixed cost ÷ contribution gross margin.
Among them, contribution gross margin = contribution gross margin/sales revenue.
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Profit and loss in quantitative relations
It is known that there are two distribution schemes, one is surplus and the other is insufficient. Find the number of people participating in the distribution and the total amount of distribution. Such problems are usually called profit and loss problems.
Knowledge background
China's ancient mathematical masterpiece "Nine Chapters Arithmetic" recorded profit and loss problem in the sixth chapter. The surplus is more than enough. Loss means deficiency.
Typical profit and loss problems are generally expressed in the following form:
Give some apples (unknown) to some people (unknown). If everyone is divided into two, there will be more than 20 apples. If everyone is divided into three, there will be five fewer apples. How many people are there in the general manager? How many apples are there?
The invariants in the topic are the number of people and the number of apples. Comparing the two different distribution methods, we can see that apples are different:
20+5 = 25 (pieces); There is no doubt that the difference of 25 apples is due to the difference of 3-2 = 1 apple per person.
In fact, there is only one situation that will lead to the above situation, that is, there are 25 people sharing apples!
After the number of people is obtained, according to the meaning of the question, the number of apples can be obtained in two ways:
2×25+20=70 (piece) or 3× 25-5 = 70 (piece).
Profit and loss problem's solution formula
The solution of one profit and one loss
(abundance+deficit) The difference between the two distributions per capita.
Double surplus solution
(Daying-Xiaoying) The difference between the two distributions per person.
Solution of double loss
(big loss-small loss), the difference between the two distributions per person.
Important and difficult
Some application questions do not seem to be profit and loss problem on the surface, but after careful analysis and appropriate conditional transformation, they can be transformed into profit and loss problem to answer.
Learning method guidance
From the formula of solving profit and loss problem, it can be seen that the key to solve this kind of problem is to carefully determine the difference between the two distributions and the total profit and loss. If the two distributions are surplus once and insufficient once, the number of people (not the number of things) is calculated according to the above formula first, and then the number of things is calculated; If both distributions are more than enough, the formula becomes the difference of surplus divided by the difference of two distributions; If both distributions are insufficient, the formula becomes the difference of the deficit divided by the difference of the two distributions. If …
Sometimes, in order to find a solution from the complex quantitative relationship, it is necessary to transform the conditions in the topic; Sometimes, it is difficult to start with "inclusion" directly, but it is easier to think indirectly from its opposite "non-inclusion".