First, the main points of knowledge

The thinking method of hypothesis decomposition is to change the conditions of the topic through hypothesis first, and then calculate with known conditions. Some problems can be thought through assumptions and ingenious solutions can be found.

When using the hypothesis method, it can be assumed that the quantity increases or decreases, which is related to the known conditions; You can also assume that the score of one quantity is the same as the score of another quantity, and then find the sum corresponding to this score according to multiplication and division and distribution, and finally solve it according to its contradiction with the actual situation.

Second, be concise.

Example 1

The sum of numbers A and B is 185. It is known that the sum of 1/4 of number A and 1/5 of number B is 42. What are these two numbers?

Mental navigation assumes "a number 1/4", "b number 1/5" and "the sum of a number 4/5 and b number is 42", and then subtracts 65438+ from 185.

Solution: b: (185-42× 4) ÷ (1-1/5× 4) = 85.

A: A's number is 100 and B's number is 85.

Exercise 1:

1. Party A and Party B * * * have the money of 150 yuan, and the sum of the money of Party A 1/2 and Party B1/0 is 35 yuan. How much money does Party A and Party B have?

2. There are 338 fire brigades A and B in total .. Team A 1/7, Team B 1/3, * * * 78. How many people are there in the two fire brigades?

Ocean Fertilizer Plant plans to produce a batch of fertilizers in the second quarter. It is known that 1/3 of the total amount completed in April is 50 tons more, 2/5 of the total amount completed in May is 70 tons less, and 420 tons are still unfinished. How many tons was originally planned to be produced in the second quarter?

Example 2

250 color TV sets and black and white TV sets. If the color TV sells 1/9, it is 5 more than the black-and-white TV. Q: How many TV sets are there in each model?

As can be seen from the figure, if five black and white TV sets are added, there will be as many as the remaining color TV sets after selling 1/9.

After adding five black and white TV sets, it is equivalent to (1- 1/9) = 8/9 for color TV sets.

(250+5) ÷ (1+1-1/9) =135 (Taiwan Province)

250-125 =115 (Taiwan Province)

A: There are 135 color TV sets and 1 15 black and white TV sets.

Exercise 2:

1. The two sisters raised 120 rabbits. If the elder sister sells 1/7, it is more than the younger sister 10. How many rabbits do my sister and sister have?

2. There are basketball and football in the school ***2 1. After lending basketball 1 3, it is less than football1. How many basketballs and footballs are there?

Jia Xiaoming has raised 65,438,000 chickens and ducks. If chickens sell 1/20, which is more than ducks 17, how many chickens and ducks are there in Xiaoming's family?

Example 3 Master and apprentice * * * processed 105 parts. It is known that the sum of 3/8 parts processed by the master and 4/7 parts processed by the apprentice is 49. How many parts do the master and apprentice process respectively?

Suppose that both master and apprentice have completed 4/7, one can complete (105× 4/7) = 60, and the difference from the actual situation (60-49) = 1 1, which is 3% of what the master has completed. In this way, we can find that the master has handled 1 1 ÷ (4/7-3/8) = 56 pieces. Namely:

Main: (105× 4/7-49) ÷ (4/7-3/8) = 56 (pieces)

Apprentice: 105-56 = 49 (pieces)

A: The master processed 56 pieces and the apprentice processed 49 pieces.

Exercise 3:

1. A store has * * 136 color TV sets and black-and-white TV sets, and sold 2/5 color TV sets and 3/7 black-and-white TV sets, and * * * sold 57 sets. Q: How many color TV sets and black-and-white TV sets were there?

2. There were 336 fire brigades A and B * *, 5/7 of which were in Team A and 3/7 of which were in Team B, and dispatched 188 people to participate in the fire fighting. Q: How many people are there in the two fire brigades A and B?

3. The school bought ***64 football and volleyball, and after lending the number of volleyball 1/4 and the number of football 1/3, there are still 46 left. How many volleyball and football did you buy?

Example 4 The sum of numbers A and B is 300, and 2/5 of number A is 55 more than 1/4 of number B. What are numbers A and B respectively?

The sum of 2/5 of a number and 2/5 of b number is 2/5 of a number and b number, that is, 300× 2/5 = 120. Because 2/5 of a number is 55 more than 1/4 of b number, the difference obtained by subtracting 55 from 120 can be regarded as b number.

b:(300×2/5-55)÷(2/5+ 1/4)= 100。

A: 300- 100 = 200.

A: The number A is 200 and the number B is 100.

Exercise 4:

1. There are 800 sheep and goats in the livestock farm, of which 2/5 are 50 more than 1/2. How many goats and sheep are there in this livestock farm?

2. The master and the apprentice * * * processed 840 parts, and 5/8 of the parts processed by the master was 60 more than 2/3 of the parts processed by the apprentice. How many parts did the master and apprentice process respectively?

3. Trees planted in Class A and Class B of Grade 6 in a school 100. The 65,438+0/65,438+00 in Class B is 65,438+06 less than that in Class A. How many trees are there in the two classes?

There were 750 students in Yuhong Primary School last term. This semester, boys increased by 1/6, while girls decreased by 1/5. * * There are 765,438+00 students. How many boys and girls are there this semester?

Suppose that the number of female students this semester has not decreased by 1/5, but increased by 1/6. There should be 750×( 1+ 1/6)= 875 students in half a semester, which is 875-765, 438+00 = 165 more than the actual number. However, the actual number of girls has decreased by 1/5, so this 165 corresponds to girls' (1/5+1/6) =11/30.

Last semester, girls: 750× (1+1/6)-710 ÷ (1/5+1/6) = 450 (person).

Female students this semester: 450× (1- 1/5) = 360 (person)

Boys this semester: 7 10-360 = 350 (person)

There are 350 boys and 360 girls this semester.

Exercise 5:

1. The weight of gold in water will decrease119, and the weight of silver in water will decrease110. A gold-silver alloy weighing 770g will weigh 720g when put in water. How many grams of gold and silver does this alloy contain?

A middle school recruited 475 freshmen last year and 640 freshmen this year. Among them, junior high school enrolled 48% more freshmen than last year, and senior high school enrolled 20% more freshmen than last year. How many freshmen are enrolled in junior high school and senior high school this year?

There are red balls and yellow balls in the bag *** 1 19. When the red ball increases by 3/8 and the yellow ball decreases by 2/5, the total number of red balls and yellow balls becomes 12 1. How many red balls and yellow balls are there in the schoolbag?

Solving Problems with Assumptions (2)

First, the main points of knowledge

Knowing that A is a fraction of B, and knowing the new multiple relationship between A and B after changing a certain number, what is the number of A and B? This kind of application problem is called variable multiple problem.

There are various situations such as two numbers increasing at the same time, two numbers decreasing at the same time, one increasing and one decreasing. Although the relationship between quantities is complicated, the key to the solution is to determine which quantity is the unit "1", and then by assuming that the difference before and after the change is equivalent to a fraction of the unit "1", so as to find the quantity of the unit "1", and other required quantities can be solved easily.

Second, be concise.

Example 1 Two iron wires, the length of the first one is three times that of the second one, and each one is 6 meters. The remaining length of the first one is five times that of the second one. How many meters is the second one?

Thinking navigation assumes that the first root is 6× 3 = 18m, so the remaining length of the first root is still three times that of the second root, but in fact, the first root is less than the assumed one by (6× 3-6) = 12m, leaving (5-3) = 2 times the remaining length of the second root.

(6× 3-3) ÷ (5-3)+6 = 12 (m)

A: The second root was originally12m.

Exercise 1:

1. The number of original books in Ding Xiao is five times that in Yang Chen. If two people lend five books to other students at the same time, the number of Ding Xiaoshu is 10 times that of Yang Chen. How many books do they have?

In terms of tree planting, the number of trees planted in Guangming Middle School is three times that in bright primary school. If 450 trees are planted in middle school and 400 trees are planted in primary school, then the trees planted in middle school are twice as many as those in primary school. How many trees were planted in primary and secondary schools?

3. Two piles of coal, the first pile is twice as heavy as the second pile, the first pile uses 8 tons, the second pile uses 1 1 ton, and the remaining weight of the first pile is four times that of the second pile. How many tons is the second pile of coal?

Example 2 Wang Ming usually saves 6.40 yuan more pocket money than Chen Gang. If two people each buy a story book of 4.40 yuan, Wang Ming's money is eight times that of Chen Gang. How much pocket money does Chen Gang have?

Assuming that Wang Ming's money is 6.40 yuan more than Chen Gang's, Wang Ming will spend 4.40× 3 = 13.20 yuan more, but Wang Ming only spent 4.40 yuan, which is less than 13.20-4.40 = 8.80 yuan 13.20 yuan. Then Wang Ming's money after buying books is 6.40+8.80 = 15.20. 3 times that of Chen Gang, and it has been said in the title that Wang Ming's money after buying books is 8 times that of Chen Gang, so 15.20 yuan corresponds to 8-3 = 5 times that of Chen Gang.

6.40+(4.40× 3-4.40÷ (8-3)+4.40 = 7.44 (Yuan)

A: Chen Gang's pocket money used to be 7.44 yuan.

Exercise 2:

1. There are three times as many books on shelf A as on shelf B. If there are 150 books on both shelves A and B, there are twice as many books on shelf A as on shelf B. How many books are there on shelves A and B respectively?

Last school year, there were 54 students in Macun Middle School, twice as many as Niuzhuang Primary School. This school year, there are 20 students in Macun Middle School and 8 students in Niuzhuang Primary School, so there are 26 students in Macun Middle School, which is four times less than that in Niuzhuang Primary School. How many students were there in Macun Middle School and Niuzhuang Primary School in the last school?

There are two kinds of glass balls, red and white. There are two more red balls than white balls and three times more than white balls. Take out 7 white balls and 15 red balls from the box at a time. After several times, there were three white balls and 53 red balls left in the box. So, how many white balls are there in the box

Example 3 The number of crayons in Xiaohong is Xiaogang's 1/2. After they bought five crayons each, Xiaohong's crayons were two-thirds of Xiaogang's. How many crayons do they each have?

Thinking navigation assumes that Xiao Hong just bought five markers and the marker is still small 1/2, so Xiao Hong only needs to buy (5× 1/2) = 2 and 1/2, but in fact Xiao Hong bought five markers and bought 5-2 and 1/2 = 2 and. When Xiao just bought five branches, the number of branches was regarded as "1", and Xiaohong bought two branches, 1/2, which was equivalent to (2/3- 1/2) = 1/6.

Xiaogang originally: (5-5×1/2) ÷ (2/3-1/2)-5 =10 (branch)

Xiaohongyuan: 10× 1/2 = 5 (branch)

A: Xiaogang used to have the 10 mark, and Xiaohong used to have five marks.

Exercise 3:

1. Xiaohua's age this year is 1/6 of his father's, and four years later, Xiaohua's age is 1/4 of his father's. How old is Obana's father this year?

Xiaohong is 3/8 of her mother's age this year. 10 years later, Xiaohong's age is her mother's 1/2. How old is Xiaohong this year?

The number of books on the A shelf is 5/7 of that on the B shelf. After adding 90 books to both the A and B shelves, the number of books on the A shelf is 4/5 of that on the B shelf. How many books are there on shelf A and shelf B respectively?

The original collection of books of Wang Fang is 4/5 of that of Li Wei. After each of them donated 10 copies to Project Hope, Wang Fang's library collection was 7/ 10 of Li Wei's. How many books do they each have?

Thinking navigation assumes that after Li Wei donated 10 books, Wang Fang's books are still 4/5 of Li Wei's, so Wang Fang only needs to donate 10×4/5 = 8, but in fact Wang Fang donated 10 books and donated 10-8 = 2 books. The remaining books after Li Wei donated books are counted. "

(10-10× 4/5) ÷ (4/5-710) = 30 (Ben)

30× 4/5 = 24 (Ben)

A: Li Wei has 30 original books and Wang Fang has 24 original books.

Exercise 4:

1.The books on shelf A are 4/5 of those on shelf B. After borrowing 1 12 from these two shelves, the books on shelf A are 4/7 of those on shelf B. How many books are there on each shelf?

Xiaoming's age this year is 6/ 1 1 of his father's. Before 10, Xiao Ming was 4/9 of his father. How old are Xiaoming and his father this year?

3. The workers in Workshop A are 1/4 of Workshop B. After 30 workers were transferred from Workshop A and Workshop B, the workers in Workshop A only accounted for 1/6 of Workshop B. How many workers were there in Workshop A and Workshop B at that time?

The number of boys in grade six in a school is 23 of that of girls. Later, two boys transferred and three girls transferred. At this time, the number of boys is 3/4 of that of girls. How many boys and girls are there now?

Thinking navigation assumes that the number of boys is still 2/3 of that of girls after transferring three girls, so boys should transfer 3 × 2/3 = 2, but in fact, boys transfer 2, with a difference of 2+2 = 4. If the number of girls after three girls transfer to another school is regarded as "1", the difference in m-girls is equivalent to 3/4-2/3 of that of girls now.

(2+3× 2/3) ÷ (3/4-2/3) = 48 (person)

48× 3/4 = 36 (person)

Now there are 36 boys and 48 girls.

Exercise 5:

1.The number of workers in workshop A is 2/5 of that in workshop B. Later, there were 20 workers in workshop A and 35 workers in workshop B, so the number of workers in workshop A was 7/9 of that in workshop B. How many people are there in workshops A and B now?

2. There is a pile of chess pieces, and the number of sunspots is 2/3 of that of Bai Zi. Now, if 12 sunspots are removed and 18 albino stars are added, the number of sunspots is 5/ 12 that of Bai Zi. How many albinos and sunspots are there now?

3. The students of Aihua Primary School and Shuguang Primary School took part in the primary school mathematics competition. In last year's competition, the number of people who won the first prize in Aihua Primary School was 2.5 times that of Shuguang Primary School. In this year's competition, the number of people who won the first prize in Aihua Primary School decreased by 1 person, and the number of people who won the first prize in Shuguang Primary School increased by 6 people. At this time, the number of people who won the first prize in Shuguang Primary School was twice that of Aihua Primary School. How many students were there in the first prize of the two universities last year?