What Is the Crisscross Method?

^[1] is a simple method for calculating the average amount and composition of two groups of mixtures. All questions that can be calculated as M1 · n1 + M2 · n2 = M · n can be calculated according to the cross method.

The calculation method of the cross method in the binary component mixed system is often used to calculate the proportional relationship of the binary components. The principle is as follows: if a, b (a> b) represent the two components in a binary component, respectively , C represents the relative average of the two components a and b, and x and y represent the proportion of a and b in the mixed system, respectively.

It's just a solution to a binary linear equation.
x + y = 1
ax + by = c
c is between a and b, solve: x: y

The cross elimination method is referred to as the cross elimination method for short. It is a solution to a class of ion inference questions. The use of "cross elimination" can narrow the scope of unknown substances, so as to use the questions to determine the conditions and find the correct answer.

The cross method is often used to calculate:

(1) Calculation of quality scores;

(2) About average

The essence of the cross method is a simple form of solving a binary linear equation. This type of problem can also be solved by equations. The specific method using this rule is as follows: the concentration of A is 10, the concentration of B is 8, and the concentration of their mixture is 9, you can put 9 in the middle, write 10 and 8 on the left, mark AB, and then subtract 9 respectively, you can get 1 and 1 on the right. The ratio is now 1: 1.

(I) Mixed Gas Calculation [Example] At normal temperature, 1 volume of ethylene is mixed with a certain amount of a certain gaseous unknown hydrocarbon. The relative density of the mixed gas to hydrogen is measured to be 12 times. Find the volume occupied by this hydrocarbon.
[Analysis] According to the calculation of the relative density, the average formula of the mixed gas is 24, and the formula of ethylene is 28, then the formula of the unknown hydrocarbon must be less than 24, and the hydrocarbon with the formula of less than 24 is only methane, which can be obtained by the cross method. Methane is 1/2 volume

(2) Calculation of atomic content [Example] There are two kinds of nuclides in bromine. In nature, these two nuclides each account for about half. It is known that the atomic number of bromine is 35 and the atomic weight is 80. The number of children is equal to each other.
(A) 79, 81 (B) 45, 46 (C) 44, 45 (D) 44, 46
[Analysis] The two isotopes account for about half of each. According to the cross method, the difference between the atomic weight of the two isotopes and the atomic weight of bromine is equal, so their neutron numbers should differ by 2, so the answer is D
Let the number of two neutrons be X and Y, respectively, because they each halve, so the following is 1: 1
X ₁
80-35 = 45
Y ₁
45 + 1 = 46, 45-1 = 44

(3) Calculation of solution preparation [Example question] A student wants to prepare 100 g of 40% NaOH solution, and 10% of NaOH solution and NaOH solids are available in the laboratory. Ask the student how many grams of the above substance should each be taken?
[Analysis] 10% NaOH solution solute mass fraction is 10%, NaOH solid "mass solute mass" is 100%, 40% NaOH solution solute mass fraction is 40%, using the cross method to obtain: 10% NaOH solution is required 2/3 × 100 = 66.7 grams, requires 1/3 × 100 = 33.3 grams of NaOH solids

(4) Calculation of mixture reaction [Example problem] The existing mixture of 100 g of lithium carbonate and barium carbonate, the amount of hydrochloric acid consumed when reacting with a certain concentration of hydrochloric acid is the same as the amount of hydrochloric acid consumed when reacting with 100 g of calcium carbonate and hydrochloric acid at that concentration. Calculate the ratio of the amounts of lithium carbonate and barium carbonate species in the mixture.
[Analysis] The formula of calcium carbonate can be understood as the average formula of a mixture of lithium carbonate and barium carbonate, and the ratio of the amount of lithium carbonate and barium carbonate can be calculated by the cross method 97:26 ^[4]

(V) Mathematical Statistics [Example 1] (2007 National Civil Service Examination Question) 7,650 students graduated from a college in 2006, an increase of 2% over the previous year. Among them, undergraduate graduates decreased by 2% compared with the previous year. And the number of graduate students has increased by 10% compared to the previous year, so how many undergraduates graduated from this university in 2006?
[Analysis] According to the cross method, the ratio of undergraduate and master can be calculated to be 2: 1.
So according to the intent, there were 7650 ÷ 1.02 = 7500 graduates in the previous year.
While undergraduate: master = 2: 1
So there were 7500 * 2/3 = 5000 undergraduates in the previous year
Undergraduates have decreased by 2% this year, so there are 5000 × 98% = 4900.