Mathematical application problems mainly include the following types of application problems: first, the concentration problem; Second, the problem of planting trees; Third, the trip problem; Fourth, the age problem; Fifth, the problem of running water; Sixth, engineering problems; Seven, the problem of proportional distribution; Eight, profit issues, etc. ; 9' price issue. Let's review these classic mathematical operation application problems again.
First, the concentration problem.
100g 70% alcohol solution mixed with 400g 20% alcohol solution, what is the concentration of alcohol solution? ( )
A.30%
B.32%
C.40%
D.45%
Analysis a. 100g 70% alcohol solution contains100× 70% = 70g;
400g of 20% alcohol solution contains 400× 20% alcohol = 80g;
Alcohol content in mixed alcohol solution = 70+80 =150g;
The total weight of the mixed alcohol solution is100+400 = 500g;
The concentration of mixed alcohol solution =150/500×100% = 30%, so choose a.
Second, the problem of planting trees.
As we all know, the circumference of planting trees around the circular flower bed is 50 meters. If you plant a tree every 5 meters, how many trees can a * * * plant? ( )
Answer 9
10
C. 1 1
D. 12
Analysis B. This question is a completely closed circular punctuation, and its number is easy to think of. That is, if a line segment is enclosed in a closed geometric figure, the starting point and the ending point coincide together, that is, there is one less point than the original, and the number of points in the unclosed figure is one more than the segmentation ratio. For example, for a line segment of ns meters, each segment of s meters is a point, so there is n+ 1. Choose B.
Third, the distance problem.
Example: It took a ship 12 hours from Port A in the upper reaches of the river to Port C in the lower reaches, then turned around and went upstream to Port B in the middle reaches. It is known that the downstream speed of this ship is twice as fast as the upstream speed. The current speed is 2 kilometers per hour, and the distance from Port A to Port B is 18 kilometers. Then the distance between port A and port C is ()
A.44 kilometers
48 kilometers
About 30 kilometers
36 kilometers in diameter
Analysis A. Downstream speed-countercurrent speed =2× water flow speed, downstream speed =2× countercurrent speed. It is known that when downstream speed =4× water flow speed = 8km/, countercurrent speed =2× water flow speed = 4km/. Let the distance from port A to port C be x kilometers, and the equation X÷8+(X- 18)÷4= 12 gives X=44. Choose a.
Fourth, the issue of age.
Dad, brother and sister are all 64 years old now. When my father was three times as old as my brother, my sister was nine years old. When my brother is twice as old as my sister, my father is 34 years old. How old is dad now? ( )
Answer 34
b39
C.40
Grass 42
Analyze C. method of substitution to solve this problem: Item A, when the father is 34 years old, the elder brother is twice as old as the elder sister, and the sum of their ages is 64-34 = 30, then when the elder brother is 20 years old, the elder sister is 10. The verification shows that when the elder sister is 9 years old, the elder brother 19 years old, and the father is 33 years old. The father's age is not three times that of the elder brother. If item A is excluded, items B and D can be excluded. Choose C.
Verb (abbreviation of verb) tap water problem
A ship is sailing on a 208-kilometer-long waterway. 8 hours downstream and 0/3 hours upstream/kloc. Find the speed of the ship in still water and the speed of the current.
0.4 km/h
B. 5 km/h
6 km/h
7 km/h
Analysis b
The speed of the ship sailing along the river is: 208÷8=26 (km/h).
The speed of sailing against the current is: 208÷ 13= 16 (km/h).
From the formula ship speed = (downstream speed+upstream speed) ÷2, we can find that the speed of this ship in still water is:
(26+ 16)÷2=2 1 (km/h)
From the formula water velocity = (downstream velocity-upstream velocity) ÷2, we can find that the water velocity is:
(26- 16)÷2=5 (km/h) Select B.
Engineering problems of intransitive verbs
Examples include two projects, A and B, which are now completed by two construction teams, A and B. In sunny days, it takes 65,438+02 days for construction team A to complete the task and 65,438+05 days for construction team B to complete it. In rainy days, the work efficiency of construction team A will be reduced by 50%, while that of construction team B will be reduced by 25%. Finally, two construction teams will start and complete these two projects at the same time. In architecture,
A .6
B.8
C.9
Number 10
Analysis A. The traditional solution of this kind of problem can be solved by a series of equations. Assuming that it is sunny for x days and rainy for y days, the equation is obtained:
X/12+y/(12× 2) = x/15+y/(15× 4/3) The result is X/Y= 1/2, that is, it is sunny.
Seven, the proportion problem
For example, there are 450 students in the first, second and third grades in a school, and the ratio of students in the three grades is 2∶3∶4. How many people are there in the grade with the largest number of students?
A. 100
B 150
C.200
A.D. 250
When solving this problem, the total number of people can be regarded as including 2+3+4=9, including 2 in Grade One, 3 in Grade Two and 4 in Grade Three. Therefore, the number of students in grade three is the largest, accounting for 4/9 of the total number, so the answer is 200. Choose C.
Eight, the profit problem
For example, if a commodity is sold at a fixed price, each of them can make a profit in 45 yuan. At present, 8 items are discounted at a fixed price, and 12 items are sold at a reduced price in 35 yuan, which can also make a profit. What's the list price of this commodity?
A. 100
B 120
C. 180
D.200
Answer and analysis D. The profit of 35 yuan sales for each price reduction is (45-35) × 12 = 120 yuan. If it is sold at a discount, the profit of each commodity is 120÷8= 15 yuan, and the loss is 45- 15 yuan = 30 yuan.
Nine price problems
Two stores, A and B, buy the same commodity. The purchase price of store A is cheaper than that of store B 10%, and the price is 20% profit. The price of store B is 15% profit, and the price of store B is higher than that of store A, so the purchase price of store A is ().
A.330 yuan B, 360 yuan C, 370 yuan D, 400 yuan
Answer b
Analysis shows that the purchase price of store A is cheaper than that of store B 10%, which must be a multiple of 9.