In the process of examination, the problems of mathematical operation module are mainly solved by substitution exclusion method, digital characteristic method, equation method and assignment method, but some complicated problems may be encountered when investigating some weighted average problems. If the conventional method equation method is adopted, it will be more difficult to formulate and solve, and the solving process will be cumbersome and consume valuable time. At this time, if you can master the cross method, the difficulty of the problem can be greatly reduced, and the result can be solved quickly. Is it a special form? Spike. .

First,? Cross method? introduce

? Cross method? Originally used to solve the problem of solution mixing, to solve the problem of solution quality or concentration before or after mixing. For example, before mixing, the mass and concentration of one solution are A and A respectively, and the mass and concentration of the other solution are B and B； respectively; The concentration of the mixed solution is R. According to the constant solute mass before and after mixing, Aa+Bb=(A+B)? R, which can be simplified as follows: the mass ratio of the solution before mixing is obtained according to the concentration data before and after mixing; To avoid this complicated simplification process, a cross can be used instead, as shown below:

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Directly available:. This method is actually a form of simplifying the equation, which conforms to Aa+Bb=(A+B)? You can use the equation form of r? Cross? To simplify the table.

Second, give detailed examples.

Example 1 The bakery bought a packet of white sugar at a price of 15 yuan/kg, and took part of it and dissolved it in water to form 20% sugar water 12 kg. Then all the remaining sugar is added and dissolved, and the sugar water concentration becomes 25%. How much did it cost to buy sugar? ( )

45 and 48

C.36 D.42

Answer b

Analysis shows that 20% white sugar water equivalent to 12kg is mixed with 100% white sugar solution to form 25% white sugar water. The cross form is as follows:

The solution is x=0.8. 12kg 20% sugar water contains 12? 20%=2.4, so sugar ***2.4+0.8=3.2kg. The cost is 3.2? 15=48 yuan. So the answer to this question is B.

Third,? Cross method? Extended application of.

? Cross method? In addition to solving the mixing problem in mathematical operation, it can also solve the average problem, discount problem, profit rate problem and mixing ratio problem; It can even be used to solve the mixed growth rate problem in data analysis. So it's called? Almighty? Type method. All the above topics have the following * * * similarities:

(1) part is related to the whole;

(2) Study the same indicators of part and whole, such as concentration, average score, discount rate, profit rate, growth rate or proportion;

(3) The ratio is the ratio of the denominator of the above indicators. The specific application process is as follows:

Example 2 A squirrel picks 24 pine nuts every day in sunny days and 16 in rainy days. It picks 168 pine nuts for several days, with an average of 2/kloc-0 pine nuts per day. How many days are there in sunny days? ( )

A.3 B.4

C.5 D.6

Answer c

The cross is analyzed as follows:

Sunny days are multiples of 5. According to multiple characteristics, there are five sunny days. So the answer to this question is C.

Example 3 A supermarket bought 65,438+0,000 watermelons, but some watermelons cracked during transportation. After the watermelons without cracks are sold out, the profit rate is 40%. The cracked watermelon can only be sold at a reduced price, with a loss of 60%. The total profit margin at the final settlement is 32%. How many watermelons have cracked? ( )

A.80

C.85 D.78

Answer a

The cross is analyzed as follows:

The cracked watermelons account for 8%/(92%+8%)= 8% of all watermelons, that is, 1000? 8%=80 (pieces). So the answer to this question is a.

The Art College of the University is divided into two departments: Music Department and Fine Arts Department. It is known that the number of male students in the college accounts for 30% of the total number of students, and the ratio of male to female students in the music department is 1:3, and that in the art department is 2: 3. What is the ratio of music department and art department to the total number of students?

A.5∶2 B.5∶ 1

C.3∶ 1 D.2∶ 1

Answer d

The essence of this problem is the mixed proportion, which crosses as follows:

That is, the ratio of the total number of music departments to the total number of fine arts departments is 2: 1. Therefore, choose the d option.

In 520 17, the per capita consumption expenditure of urban and rural residents in a certain area was 37,425 yuan, up 6.4% year-on-year. Among them, the per capita consumption expenditure of urban residents was 40,346 yuan, a year-on-year increase of 5.5%; Per capita consumption expenditure of rural residents 188 10 yuan, up 8.5% year-on-year.

In 20 17, how many times more urban residents in this area than rural residents? ( )

A.5.2 B.6.4

C.6.9 D.7.8

Answer b

Analysis of urban and rural = urban+rural, when the average of the two parts and the average of the whole are known, the ratio of the number of people in the two parts can be calculated by the cross method, and the data bits are more. After the same treatment, the cross form is as follows:

So the answer to this question is B.

Example 6 (Shanxi 2014-16) In 2012 years, China's foreign trade in mineral products was active, with the total import and export volume of 9.91900 million US dollars, up 3.6% year-on-year, of which imports increased1.

20 1 1 year, what is the total import volume of mineral products in China? ( )

A. 1.5

C.2. 1 D.2.5

Answer b

Analysis of 20 12 years import growth rate 1.4%, export growth rate 7.6%, import and export growth rate 3.6%. The cross form is as follows:

It can be concluded that the ratio of import value to export value of 20 1 1 is 4%: 2.2%? 1.8。 So the answer to this question is B.

Note: this question is about the ratio of base period quantity, and the equivalent relationship is Aa+Bb=(A+B)? R, the substitute data is: 20 1 1 annual import amount? 1.4%+20 1 1 annual export volume? 7.6% = 20 1 1 annual total import and export? 3.6%。

Fourth, matters needing attention

? Cross method? Pay attention to the following points when applying:

(1) used to quickly solve the problem of the proportion of the two parts;

② The mixed data is centered, and the diagonal line should be the result of restoring data from big data;

③ Different types of questions should be accurately distinguished from A; When it comes to growth rate, both A and B represent the base period.