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The first day of the first volume)
First, the first quality control:
1. Warriors volleyball team scored 1:3, 3:2, 0:3, 3: 1. What's the total net win? P6 Page
Solution: 1+3+3-(3+2+3+ 1)
=7-9
=-2
A: The total net win is -2.
2. The following figures are the math test scores of 10 students. Estimate their average score first, and then calculate their average score on this basis to test your valuation ability. P6 Page
82, 83, 78, 66, 95, 75, 56, 93, 82, 8 1
I estimate that their average score is 80 points.
Solution: (82+83+78+66+95+75+56+93+82+81) present10.
=79 1÷ 10
=79. 1 (minimum)
A: Their average grade is 79. 1.
3. When the temperature rises by 65438±0°C, a wire will be extended by 0.002 mm. On the contrary, when the temperature drops1C, the wire will be shortened by 0.002 mm.15 C. What happens to the length of the wire when it is heated to 60°C and then cooled to 5 C? How much longer is the final length than the original length? Page P7
Solution: (1), (60- 15)×0.002=0.09(mm)
⑵、0.09-(60-5) ×0.002
=0.09-0. 1 1
=-0.02 mm
Answer: The final length is -0.02mm longer than the original length.
The distance between the earth and the sun changes with time in a year. 1 astronomical unit is the average distance from the earth to the sun, that is, 65,438+49.6 million kilometers. Try to express how many kilometers 1 astronomical unit is in scientific notation (keep 4 significant figures). Page P7
Solution: 1.4960 (100 million kilometers), 4 significant digits.
≈ 1.496× 108 (km)
An astronomical unit is about 1.496× 108 km.
Inequalities and Inequality Groups (Application Questions)
A shop bought 200 bicycles at the purchase price of each 250 yuan, and sold them at the price of 275 yuan each. Two months later, the sales amount of bicycles has exceeded the purchase amount of these bicycles. How many bicycles have been sold at least at this time? P54 page
Solution: Suppose at least X bicycles are sold at this time.
275 times; 250×200
275 times; 50000
x﹥ 18 1. 1 1 ......
X is an integer.
∴ X= 182
A: At least 65,438+082 bicycles were sold.
6. When the quarry is blasted, the workers should ignite the fuse and move to a safe area 400 meters away before blasting. The burning speed of fuse is 1 cm/s, and the transfer speed of workers is 5 m/s. How long does it take to fuse?
Solution: it takes at least x meters to set the fuse.
400÷5≤X/0.0 1
80≤X/0.0 1
X≥0.8
Answer: The fuse should be at least 0.8m. ..
7. It takes 10 hour for a ship to sail from A in the upper reaches of A River to B in the lower reaches, and it takes less than 12 hours to return from B at a constant speed. The speed of this river is 3 km/h, and the still water speed of the ship is V.
Constant, what conditions does v meet? P54 page
Solution: Let the velocity of still water be V, and we get
(3+V)× 10 \( V-3) 10
(3+V)× 10 \( V-3)÷ 12
Solution: v-33
Answer: Static speed V-33
◆8. The purchase price of apples is 1.5 yuan per kilogram, and it is estimated that 5% of apples are normally lost in sales. What is the lowest price that a merchant can set so as not to lose money? P54 page
Solution: Suppose the merchant sets the price at least X yuan.
1.5≤( 100%-5%)X
1.5≤0.95X
X≥ 1.5789
A: If the merchants set the price at least 1.58 yuan, they can avoid losses.
9. There were 280 people in a factory the year before last, but 40 people were reduced after the structural reform last year. The annual profit of the whole plant will increase by at least 6,543,800 yuan, and the per capita profit will increase by at least 6,000 yuan. What was the profit of the whole factory the year before last?
Solution: Let's assume that the profit of the whole factory in the previous year was X million yuan. P55 page
X÷280+0.6÷(X+ 100)÷(280-40)
6X+ 1008￠7(X+ 100)
-X ￠- 1008+ 100
-X ￠- 308
X﹥308
A: The profit of the whole factory in the previous year was 3.08 million yuan.
◆ 10.2002, the proportion of days with excellent air quality (above Grade II) in Beijing reached 55%. If the proportion exceeds 70% by 2008, how many days with good air quality will increase in 2008 at least compared with 2002? (365 days per year) P55 pages
Solution: Suppose that the number of days with good air quality in 2008 is at least X days more than that in 2002.
X≥365×(70%-55%)
X≥365× 15%
X≥54.75
A: The number of days with good air quality in 2008 was at least 55 days more than that in 2002.
1 1. has a two-digit number. If its single digit A and ten digit B are reversed, under what circumstances will the two digits be larger than the original two digits? When do you get two digits smaller than the original two digits? When did you get two digits equal to the original two digits? P55 page
Solution:10a+b-10b+a (1)
10b+a； 10a+b(2)
10a+b= 10b+a (3)
a﹥b ( 1)
B; A (2)
a =b (3)
∴ (1), when a b, the obtained two digits are greater than the original two digits.
(2) when B ~ A, the obtained two-digit number is smaller than the original two-digit number.
(3) When b=a, the obtained two-digit number is equal to the original two-digit number.
12. There are 20 questions in a knowledge contest. If each question is answered correctly, it will get 10, and if it is answered incorrectly, 5 points will be deducted. Xiaoming got more than 90 points in the exam. He should at least answer a few questions correctly. P55 page
Solution: Suppose he got at least X questions right.
10X-(20-X)×5； 90
10X- 100+5X； 90
15X； 190
X﹥ 12.66……
X is an integer.
∴X= 13
A: He must answer at least 13 questions correctly.
13. A piece of gold and silver jewelry weighs one gram. According to merchants, the gold content is not less than 90%, and the densities of gold and silver are 19.3g/cm3 and 10.5g/cm3 respectively. List inequalities and explain what conditions the volume of this kind of jewelry should meet. P56 page
(Hint: mass = density × volume)
Solution: v ÷ 0.9a ÷19.3+0.1a ÷10.5.
◆14. Two stores, A and B, sell the same goods at the same price, and each offers different preferential schemes: after purchasing 100 yuan in store A, the re-purchased goods will be charged at 90% of the original price; After the accumulative purchase of 50 yuan goods in Store B, the re-purchased goods will be charged at 95% of the original price. How can customers choose a store to get a bigger discount? P56 page
Solution: let the customer's consumption amount be X yuan.
A 100+(X- 100)×0.9
B 50+(X-50)×0.95
A-B
∴ 100+(x- 100)×0.9￠50+(x-50)×0.95
x \u 150
For example, the X-50 is not discounted at stores A and B..
When the price is 50-x- 100, buy a discount at the second store.
When it is 100￠X 655438+050, buy a discount at the second store.
X- 150, go to the store to buy a discount.
15, an English book ***98 pages, Zhang Li didn't finish it for a week (7 days), but Li Yong finished it in less than a week. On average, Li Yong reads three more pages a day than Zhang Li. How many pages does Zhang Li read every day (the answer is an integer)? Page 60
Solution: Let Li Yong read (X+3) pages every day, and Zhang Li read x pages every day.
7X \u 98( 1)
7(X+3)98(2)
x \u 14( 1)
X﹥ 1 1 (2)
The solution set of inequality is 1 1￠X￠ 14.
X is an integer.
∴ X= 12， 13
Answer: I read 12 and 13 pages every day on average.
16, three teams plan to produce 500 products on 10 (the daily output is unchanged), and the task cannot be completed at the original production speed; If each team produces more 1 product every day than before, it can finish the task ahead of schedule. How many products did each group initially produce every day? Page 60
Solution: Suppose that each group originally produced X products every day.
3x× 10÷500( 1)
3(X+ 1)× 10﹥500 (2)
x￠50/3( 1)
X﹥47/3 (2)
∴ 47/March x 50/March
X is an integer.
∴ X= 16
A: Each group initially produced 65,438+06 products per day.
17. The price of a commodity is 150 yuan, and the profit that a merchant can get from selling a commodity is 10% ~ 20% of the purchase price. What is the range of purchase price (accurate to 1 yuan)? Page 62
Solution: Let the purchase price be X yuan.
X+ 10%X= 150 ( 1)
X+20%X= 150 (2)
X≈ 136 ( 1)
X= 125 (2)
The purchase price range of ∴ is 125 yuan ~ 136 yuan.
◆ 18, pumping with a type A pump that can pump 1. 1 ton of water per minute can be finished in half an hour; If B-type pump is used, it is estimated that it will take 20 to 22 minutes to finish pumping. How many tons of water does type B pump pump per minute more than type A pump? Page 63
Solution: Type B pump can pump X tons of water per minute.
20≤ 1. 1×30/X≤22
20X≤ 1. 1×30
22X≥ 1. 1×30
20X≤33
22X≥33
X≤ 1.65
X≥ 1.5
∴ 1.5≤X≤ 1.65
1.5- 1. 1=0.4
1.65- 1. 1=0.55
Set the type B pump to pump y tons more water per minute than the type A pump.
∴0.4≤Y≤0.55
A: How much 0.4 ~ 0.55 tons of water is pumped per minute by the B-type pump than by the A-type pump?
◆ 19. Distribute some books to several students. If each student is divided into three books, then the remaining eight books; If each student in front gets 5 copies, then the last student gets less than 3 copies. How many books are there? How many students are there? Page 64
Solution: Suppose there are X books and Y students.
3Y+8=X ( 1)
5(Y- 1)+3=X (2)
Solution: 3Y+8=X (1)
5Y-X =2 (2)
(2)-( 1) gives 2Y= 10.
Y=5
Substitute Y=5 (1).
15+8=X
X=23
∴ X=23
Y=5
There are 23 of these books? There are five students?
Solving application problems with column equations
1. To transport 29.5 tons of coal, transport it three times by a truck with a load of 4 tons, and transport the rest by a truck with a load of 2.5 tons. How many times will it take to complete?
Solution: suppose it takes x times to complete.
29.5-3×4 = 2.5 times
17.5=2.5x
x=7
A: It will take seven more shipments to complete.
2. The trapezoidal site covers an area of 90 square meters, with an upper bottom of 7 meters and a lower bottom of 1 1 meter. How high is it?
Solution: Its height is x meters.
x(7+ 1 1)=90*2
18x= 180
x= 10
Its height is 10 meter.
A workshop plans to produce 5480 parts in April. It has been in production for 9 days, and 908 more are about to complete the production plan. How many pieces will be produced on average every day in these 9 days?
During these 9 days, X was produced every day on average.
9x+908=5408
9x=4500
x=500
During these nine days, an average of 500 were produced every day.
4. Two cars, Party A and Party B, are driving in opposite directions from two places 272km apart. Three hours later, the distance between the two cars is still 17km. A travels 45 kilometers per hour, and B travels how many kilometers per hour?
B travels x kilometers per hour.
3(45+x)+ 17=272
3(45+x)=255
45+x=85
x=40
B is driving at a speed of 40 kilometers per hour.
5. There are two classes in grade six in a school, and the average score of mathematics last semester is 85. There are 40 students in six classes (1) with an average score of 87. 1. There are 42 students in Class Six (2). What is the average score?
The average score is X.
40*87. 1+42x=85*82
3484+42x=6970
42x=3486
x=83
The average score is 83 points.
6. The school bought 10 boxes of chalk. 250 boxes were used, and there are 550 boxes left. How many boxes per box?
Average x boxes per box
10x=250+550
10x=800
x=80
80 boxes per box on average.
There are 200 students in the fourth grade. During extracurricular activities, 80 girls went to jump rope. The boys are divided into five groups to play football. How many people are in each group?
There are x people in each group on average.
5x+80=200
5x= 160
x=32
There are 32 people in each group on average.
8. The canteen brought 150 kg of rice, which was 30 kg less than the flour. How many kilograms of flour does the canteen send?
The canteen brought x kilograms of flour.
3x-30= 150
3x= 180
x=60
The canteen brought 60 kilograms of flour.
9. There are 52 peach trees and 6 rows of pear trees in the orchard. There are 20 more pear trees than peach trees. How many pear trees are there in each row on average?
There are x pear trees per row on average.
6x-52=20
6x=72
x= 12
There are 12 pear trees in each row on average.
10, the area of a triangle is 840 square meters, the base is 140 meters, and the height is how many meters?
The height is x meters.
140x=840*2
140x= 1680
x= 12
The height is12m.
1 1, Li Shifu bought 72 meters of cloth to make 20 clothes for adults and 16 clothes for children. Adult dresses are 2.4 meters each, and how many meters are children's dresses?
Each piece of children's clothing is made of x meters.
16x+20*2.4=72
16x=72-48
16x=24
x= 1.5
Each piece of children's clothing1.5m.
12, three years ago, my mother was six times older than my daughter. This year, my mother is 33 years old. How old is my daughter?
My daughter is x years old this year.
30=6(x-3)
6x- 18=30
6x=48
x=8
My daughter is 8 years old this year.
13. How long will it take for a car with a speed of 50 kilometers per hour to catch up with a car with a speed of 40 kilometers per hour that left two hours ago?
It takes x time
50x=40x+80
10x=80
x=8
It takes eight hours.
14. Xiaodong bought 3 Jin of apples and 2 Jin of pears at the fruit shop and paid 15 yuan. 1kg apples are more expensive than 1kg pears in 0.5 yuan. How much are apples and pears per kilogram?
Apple x
3x+2(x-0.5)= 15
5x= 16
x=3.2
Apple: 3.2
Pear: 2.7
15, A and B start from A and B at the same time, respectively, in opposite directions. A travels at a speed of 50 kilometers per hour, and B travels at a speed of 40 kilometers per hour. A reaches the midpoint 1 hour earlier than B. How many hours does it take for A to reach the midpoint?
X hours to reach the midpoint.
50x=40(x+ 1)
10x=40
x=4
A reaches the midpoint in 4 hours.
16, Party A and Party B set out from A and B at the same time, face to face, and meet two hours later. If A starts from place A and B goes in the same direction at the same time, then A will catch up with B after 4 hours. Given that the speed of A is 15km/, find the speed of B. ..
Speed x of b
2(x+ 15)+4x=60
2x+30+4x=60
6x=30
x=5
B's speed 5
17. Two ropes with the same length, the first one was cut15m, and the second one was 3m more than the first one. How long are these two ropes?
It turns out that these two ropes are each x meters long.
3(x- 15)+3=x
3x-45+3=x
2x=42
x=2 1
It turns out that the two ropes are each 2 1 m long.
18. A school bought 7 basketballs and 10 soccer balls and paid 248 yuan. As we all know, the price of each basketball is equivalent to the price of three footballs. How much is basketball and football each?
Every basketball x
7x+ 10x/3=248
2 1x+ 10x=744
3 1x=744
x=24
Each basketball: 24
Each football: 8
1. To transport 29.5 tons of coal, transport it three times by a truck with a load of 4 tons, and transport the rest by a truck with a load of 2.5 tons. How many times will it take to complete?
It takes x times to complete.
29.5-3 * 4 = 2.5 times
17.5=2.5x
x=7
Seven more shipments are needed to complete.
2. The trapezoidal site covers an area of 90 square meters, with an upper bottom of 7 meters and a lower bottom of 1 1 meter. How high is it?
Its height is x meters.
x(7+ 1 1)=90*2
18x= 180
x= 10
Its height is 10 meter.
A workshop plans to produce 5480 parts in April. It has been in production for 9 days, and 908 more are about to complete the production plan. How many pieces will be produced on average every day in these 9 days?
During these 9 days, X was produced every day on average.
9x+908=5408
9x=4500
x=500
During these nine days, an average of 500 were produced every day.
4. Two cars, Party A and Party B, are driving in opposite directions from two places 272km apart. Three hours later, the distance between the two cars is still 17km. A travels 45 kilometers per hour, and B travels how many kilometers per hour?
B travels x kilometers per hour.
3(45+x)+ 17=272
3(45+x)=255
45+x=85
x=40
B is driving at a speed of 40 kilometers per hour.
5. There are two classes in grade six in a school, and the average score of mathematics last semester is 85. There are 40 students in six classes (1) with an average score of 87. 1. There are 42 students in Class Six (2). What is the average score?
The average score is X.
40*87. 1+42x=85*82
3484+42x=6970
42x=3486
x=83
The average score is 83 points.
6. The school bought 10 boxes of chalk. 250 boxes were used, and there are 550 boxes left. How many boxes per box?
Average x boxes per box
10x=250+550
10x=800
x=80
80 boxes per box on average.
There are 200 students in the fourth grade. During extracurricular activities, 80 girls went to jump rope. The boys are divided into five groups to play football. How many people are in each group?
There are x people in each group on average.
5x+80=200
5x= 160
x=32
There are 32 people in each group on average.
8. The canteen brought 150 kg of rice, which was 30 kg less than the flour. How many kilograms of flour does the canteen send?
The canteen brought x kilograms of flour.
3x-30= 150
3x= 180
x=60
The canteen brought 60 kilograms of flour.
9. There are 52 peach trees and 6 rows of pear trees in the orchard. There are 20 more pear trees than peach trees. How many pear trees are there in each row on average?
There are x pear trees per row on average.
6x-52=20
6x=72
x= 12
There are 12 pear trees in each row on average.
10, the area of a triangle is 840 square meters, the base is 140 meters, and the height is how many meters?
The height is x meters.
140x=840*2
140x= 1680
x= 12
The height is12m.
1 1, Li Shifu bought 72 meters of cloth to make 20 clothes for adults and 16 clothes for children. Adult dresses are 2.4 meters each, and how many meters are children's dresses?
Each piece of children's clothing is made of x meters.
16x+20*2.4=72
16x=72-48
16x=24
x= 1.5
Each piece of children's clothing1.5m.
12, three years ago, my mother was six times older than my daughter. This year, my mother is 33 years old. How old is my daughter?
My daughter is x years old this year.
30=6(x-3)
6x- 18=30
6x=48
x=8
My daughter is 8 years old this year.
13. How long will it take for a car with a speed of 50 kilometers per hour to catch up with a car with a speed of 40 kilometers per hour that left two hours ago?
It takes x time
50x=40x+80
10x=80
x=8
It takes eight hours.
14. Xiaodong bought 3 Jin of apples and 2 Jin of pears at the fruit shop and paid 15 yuan. 1kg apples are more expensive than 1kg pears in 0.5 yuan. How much are apples and pears per kilogram?
Apple x
3x+2(x-0.5)= 15
5x= 16
x=3.2
Apple: 3.2
Pear: 2.7
15, A and B start from A and B at the same time, respectively, in opposite directions. A travels at a speed of 50 kilometers per hour, and B travels at a speed of 40 kilometers per hour. A reaches the midpoint 1 hour earlier than B. How many hours does it take for A to reach the midpoint?
X hours to reach the midpoint.
50x=40(x+ 1)
10x=40
x=4
A reaches the midpoint in 4 hours.
16, Party A and Party B set out from A and B at the same time, face to face, and meet two hours later. If A starts from place A and B goes in the same direction at the same time, then A will catch up with B after 4 hours. Given that the speed of A is 15km/, find the speed of B. ..
Speed x of b
2(x+ 15)+4x=60
2x+30+4x=60
6x=30
x=5
B's speed 5
17. Two ropes with the same length, the first one was cut15m, and the second one was 3m more than the first one. How long are these two ropes?
It turns out that these two ropes are each x meters long.
3(x- 15)+3=x
3x-45+3=x
2x=42
x=2 1
It turns out that the two ropes are each 2 1 m long.
18. A school bought 7 basketballs and 10 soccer balls and paid 248 yuan. As we all know, the price of each basketball is equivalent to the price of three footballs. How much is basketball and football each?
Every basketball x
7x+ 10x/3=248
2 1x+ 10x=744
3 1x=744
x=24
Each basketball: 24
Each football: 8
1, transporting a batch of goods, we have rented these two large trucks twice: for the first time, 2 trucks of Class A and 3 trucks of Class B transported 15.5 tons. The second time, there were 5 vehicles in Class A and 6 vehicles in Class B, and the cargo was transported 35 tons. Now, I rent three Class A cars and five Class B cars from the company. If the freight is paid per ton, 30 yuan asks the owner how much he should pay.
Solution: If A can hold X tons and B can hold Y tons, then
2x+3y= 15.5
5x+6y=35
Get x=4.
y=2.5
Get (3x+5y)*30=735。
2. A commodity is now reduced in price 10% for promotion. In order to keep the total sales amount unchanged, how much is the sales volume increased over the original price?
Solution: Increase X% when selling at the original price.
( 1- 10%)*( 1+X %)= 1
x % = 1 1. 1 1%
In order to keep the total sales amount unchanged, the sales volume increased by11.11%compared with the original price.
3. 10% reduced 1 product price is just over half of the original price. 40 yuan, what is the original price of this product?
Solution: Let the original price be X yuan.
( 1- 10%)x-40=0.5x
x= 100
A: original price 100 yuan.
4. There is 40 grams of salt water with 8% salt. How many grams of salt does it take to make salt water with a salt content of 20%?
Solution: suppose x grams of salt is added.
At first, the pure salt was 40 * 8% g.
Add x grams to make 40 * 8%+X.
Salt water is 40+x grams.
Concentration 20%
So (40*8%+x)/(40+x)=20%.
(3.2+x)/(40+x)=0.2
3.2+x=8+0.2x
0.8x=4.8
x=6
So add 6 grams of salt.
In the market, the sales of eggs are priced according to the quantity. A vendor bought a batch of eggs at a price of 0.24 yuan each, but accidentally broke 12 eggs in the process of selling. The remaining eggs are sold at 0.28 yuan each, and the result is still 1 1.2 yuan's profit. How many eggs did the vendor buy at first?
Solution: Suppose the vendor bought X eggs at first.
List the equations according to the meaning of the question:
(X- 12)* 0.28-0.24 X = 1 1.2
0.28 x-3.36-0.24 x = 1 1.2
0.04X= 14.56
X=364
A: The vendor bought 364 eggs at first.
6. A workshop has 85 technicians. On average, each person can process 15 pieces of A or 10 pieces of B every day, and two pieces of A and three pieces of B are matched with one set. How many people are arranged to process parts A and B respectively to make the two parts processed every day just match?
Solution: Suppose it takes X people to arrange the production of A, then there are (85-x) people to produce B.
Because two A-type parts and three B-type parts are matched with one set, so
Therefore, the number of parts A multiplied by 3 can be equal to the number of parts B multiplied by 2.
16*x*3= 10*(85-x)*2
Solution: x=25
It takes 25 people to produce A and 60 people to produce B!
7. Hong Guang Electric Company can still make a profit of 20% by selling a certain color TV at 20% of the marked price. It is understood that the purchase price of this color TV is 1996 yuan. So what should the price of this color TV be?
Solution: Set the price to X yuan.
80%X= 1996×( 1+20%)
80%X= 2395.2
X=2994
8. A store can make a profit of 20% by selling a commodity at 20% of the marked price. If the purchase price of this commodity is per piece in 22 yuan, what is the list price of each commodity?
Solution: Set the price to X yuan.
80%X=22×( 1+20%)
80%X= 26.4
X=33
9. On a double-track railway, two trains passed head-on. The speed of train A is 20m/s and the speed of train B is 24m/s. If the total length of train A is 180m and the total length of train B is 160m, how many seconds will it take for the two trains to miss each other?
Solution: (180+160)/(20+24) = 7.28 seconds.
10, two students, A and B, walked in the same direction from two places 5 kilometers apart on the same road. Party A's speed is 5km/h, Party B's speed is 3 km/h, and Party A takes the dog. When Party A chases Party B, the dog chases Party B first, then returns to meet Party A, and then returns to chase Party B. It is known that the speed of the dog is 15km/h, and the total distance that the dog runs in this process is calculated.
Solution: First of all, it must be clear that the meeting time between Party A and Party B is equal to the running time of the dog.
So the dog's time = A/B meeting time = the sum of total distance /A/B speed.
=5km/(5km/h+3km/h) = 5/8h
So the distance of the dog = the time of the dog * the speed of the dog =5/8h* 15km/h=75/8km.
So Party A and Party B met and the dog walked 75/8 kilometers.
One day, Xiaohong and Liang Xiao used the temperature difference to measure the height of the mountain peak. The temperature of Xiaohong at the top of the mountain is-1 degree. The temperature in Liang Xiao is 5 degrees at the foot of the mountain. This place is well known.
Every time the height in this area increases by 100M, the temperature drops by about 0.6 degrees. How high is this mountain?
When the temperature rises by 65438 0 degrees, a metal wire will be extended by 0.002MM; conversely, when the temperature drops by 65438 0 degrees, the metal wire will be shortened by 0.002 mm. 15 degrees. When the wire is heated to 60 degrees and cooled to 5 degrees, what happens to the length of the wire? How much longer is the final length than the original length?