The total number of plants is 900+ 1250 = 2 150, and 24+30+32 = 86 plants can be planted every day.
The planting days are 2 150 ÷ 86 = 25 days.
24× 25 = 600 trees will be completed in 25 days.
Then B will finish 900-600=300 trees before helping C.
That is, after 300 ÷ 30 = 10 days, I transferred from A to B on 1 1 day.
2. There are three grasslands with an area of 5 15 and 24 mu respectively. The grass on the grassland is as thick and grows as fast. The first grassland can feed 10 cows for 30 days, and the second grassland can feed 28 cows for 45 days. How many cows can eat on the third grass for 80 days?
This is a complicated problem of herding cattle.
The grass eaten by each cow every day is 1 serving.
Because the first piece of grassland with an area of 5 mu+grassland with an area of 5 mu = 10× 30 = 300 copies for 30 days.
Therefore, the amount of grass per mu and the amount of grass per mu for 30 days are 300 ÷ 5 = 60.
Because the original grass quantity of the second grassland with an area of 15 mu+the grass quantity of 65438 with an area of 45 days = 28× 45 = 1260.
Therefore, the amount of raw grass per mu and the amount of grass in 45 days per mu are 1260 ÷ 15 = 84 copies.
So 45-30 = 15 days, and the area per mu is 84-60 = 24.
Therefore, the area per mu is 24/ 15 = 1.6 parts/day.
Therefore, the amount of grass per mu is 60-30× 1.6 = 12.
The third plot covers an area of 24 mu, and needs to grow 1.6× 24 = 38.4 pieces every day, and the original grass has 24× 12 = 288 pieces.
Every day, 38.4 cows need to eat the newly grown cows, and the remaining cows eat the original grass every day, so the original grass is enough for 80 days, so 288 ÷ 80 = 3.6 cows.
So a * * * needs 38.4+3.6 = 42 cows to eat.
Two solutions:
Solution 1:
Assume that the daily grazing amount of each cow is 1, and the total grass amount per mu for 30 days is10 * 30/5 = 60; The total grass yield per mu in 45 days is: 28*45/ 15=84, so the new grass yield per mu per day is (84-60)/(45-30)= 1.6, and the original grass yield per mu is 60-1.6 *.
Scheme 2: 10 cows eat 5 mu in 30 days, and 30 cows eat 5 mu in 30 days 15 mu. According to 28 cows eating15mu for 45 days, it can be deduced that15mu of new grass (28 * 45-30 * 30)/(45-30) = original grass amount15mu:1260-24 *. 15mu cattle required for 80 days 180/80+24 (head) 24mu: (180/80+24) * (24/15) = 42 heads.
3. A project, contracted by both parties, can be completed in 2.4 days and needs to be paid 1800 yuan; Contracted by Team B and Team C, it can be completed in 3+3/4 days, and it needs to be paid 1500 yuan; Contracted by two teams, Party A, Party B and Party C, it can be completed in 2+6/7 days at a cost of 1.600 yuan. On the premise of ensuring the completion within one week, which team will spend the least?
The cooperation between Party A and Party B is completed in one day 1 ÷ 2.4 = 5/ 12, and the payment 1800 ÷ 2.4 = 750 yuan.
The cooperation between ethylene, propylene and Fang Yitian is 1 ÷ (3+3/4) = 4/ 15, and the payment is 1500× 4/ 15 = 400 yuan.
The cooperation between Party A, Party C and Fang Yitian is 1÷ (2+6/7) = 7/20, and the payment is 1600× 7/20 = 560 yuan.
Three people cooperate in one day (5/12+4/15+7/20) ÷ 2 = 31/60,
Three people cooperate to pay (750+400+560) ÷ 2 = 855 yuan a day.
Party A alone completes 31/60-4/15 =1/4 every day, and pays 855-400 = 455 yuan.
Party B alone completes 3 1/60-7/20 = 1/6 every day, and pays 855-560 = 295 yuan.
Party C alone completes 31/60-5/12 =110 every day and pays 855-750 = 105 yuan.
So by contrast,
Choose b with 1 ÷ 1/6 = 6 days, only 295× 6 = 1770 yuan.
There is a rectangular iron block in the cylindrical container. Now turn on the tap and pour the water into the container. In 3 minutes, the water surface is just above the top of the cuboid. /kloc-After 0/8 minutes, the container has been filled with water. It is known that the height of a container is 50 cm and the height of a cuboid is 20 cm. Find the ratio of the bottom area of a cuboid to the bottom area of a container.
Divide this container into upper and lower parts. According to the time relationship, it can be found that the volume of water in the upper part is 18 ÷ 3 = 6 times that of the lower part.
The height ratio of the upper half and the lower half is (50-20): 20 = 3: 2.
So the bottom area of the upper part is 6 ÷ 3× 2 = 4 times the bottom area of the lower part filled with water.
Therefore, the ratio of the bottom area of the cuboid to the bottom area of the container is (4- 1): 4 = 3: 4.
Unique solution:
(50-20): 20 = 3: 2, and when there is no cuboid, it takes 18*2/3= 12 (minutes) to fill 20 cm.
So the volume of a cuboid is 12-3=9 minutes of water, because the height is the same.
So the volume ratio is equal to the bottom area ratio, 9: 12 = 3: 4.
Two bosses, A and B, bought a fashion at the same price, and B bought 1/5 sets more than A, and then they sold them at profit margins of 80% and 50% respectively. After both of them were sold out, A still made more profits than B, just enough for him to buy 65,438+00 sets of this fashion, which A bought at the beginning.
Consider that the number of groups of A is 5 and the number of groups of B is 6.
The profit earned by A is 80% × 5 = 4, and the profit earned by B is 50% × 6 = 3.
A is 4-3 more than B = 1 copy, and this1copy is 10 set.
So, A initially purchased 10× 5 = 50 sets.
Example 1, (to solve the practical problem of "how many percent is one number more than another")
Xiangyang Bus Factory originally planned to produce 5,000 buses, but actually produced 5,500 buses. What is the percentage of actual output exceeding the plan?
Analysis and solution: The requirement of "how many percent of the actual production is more than the planned production" means that the number of vehicles whose actual production exceeds the planned production accounts for how many percent of the planned production, and the original planned production is regarded as "1". The relationship between them can be represented by a line graph.
designed output
5,000 vehicles are actually more than planned.
actual output
5,500 cars
Answer: Method 1:
5500–5000 = 500 (vehicles) ... 500 cars were actually produced more than planned.
500 ÷ 5000 = 0. 1 = 10% ... What is the percentage of actual output exceeding the plan?
Method 2:
5500 ÷ 5000 = 1 10% ... The actual output is equivalent to the original planned 1 10%.
110%-100% =10% ... What is the percentage of actual output exceeding the plan?
A: Actually, the production is more than planned 10%.
Example 2, (solving the practical problem of "how many percent is one number less than another")
Xiangyang Bus Factory originally planned to produce 5,000 buses, but actually produced 5,500 buses. What percentage is the planned output less than the actual output?
Analysis and solution: It is required that "the planned output is less than the actual output", that is, the percentage of the planned output of vehicles is less than the actual output, and the actual output is taken as the unit "1". The relationship between them can be represented by a line graph.
designed output
5,000 cars
The plan is less than the reality.
actual output
5,500 cars
Answer: Method 1:
5500–5000 = 500 (vehicles) ... It is planned to produce 500 vehicles less than the actual ones.
500 ÷ 5500 ≈ 9. 1% ... What percentage is the planned output less than the actual output?
Method 2:
5500 ÷ 5500 ≈ 90.9% ... The planned output is equivalent to the actual 90.9%.
100%-90.9% ≈ 9. 1% ... What percentage is the planned output less than the actual output?
A: The planned output is 9. 1% less than the actual output.
Comments: Think about the most basic quantitative relationship in the application problem of fractional multiplication: "unit 1 × fraction = the quantity corresponding to the fraction". If the problem is applied in combination with percentages, how much percentage one quantity is more (less) than another quantity is actually a score. Just use "more (less) units 1".
Example 3, (Breakthrough of Difficulties)
A basket of apples is 20% heavier than a basket of pears, so a basket of pears is 20% lighter than a basket of apples.
Analysis and solution: apples are 20% heavier than pears, that is, the part of apples that is heavier than pears accounts for 20% of pears, and the quality of pears is regarded as the unit "1"; The fact that pears are 20% lighter than apples means that the parts of pears that are lighter than apples account for 20% of apples. The quality of apples is regarded as the unit "1", and the two units "1" are different. It is forbidden to confuse the two issues. A basket of apples is 20% heavier than a basket of pears, which is regarded as "1". Pear is 100, and apple is 100+20 = 120. What percentage is a basket of pears lighter than a basket of apples = the part of a basket of pears lighter than a basket of apples ÷ apples = (120-100) ÷120 ≈16.7%.
A: A basket of apples is 20% heavier than a basket of pears, so a basket of pears is lighter than a basket of apples 16.7%.
Comments: In the application problem of finding the percentage of one number more (less) than another number, the key is to find the quantity of the unit "1". From the conclusion, it can be concluded that "one number is a few percent more than the other, and the other number is a few percent less than the other." This sentence is wrong. Why? Comparing the two percentages, it can be concluded that the amount corresponding to these two percentages is greater than one or less than the other, and these two statements are the same, indicating the same amount; The unit "1" is a pear and an apple, so these two percentages can't be equal.
Example 4, (Angle of View of Test Point)
An electronic product, the original price of 5000 yuan a set, is now reduced to 3000 yuan. How much is the price reduction?
Analysis and solution: reduce it to 3000 yuan, that is, the current price is 3000 yuan, that is, reduce it by 2000 yuan. A few percent reduction is a few percent of the original price.
5000–3000 = 2000 (yuan)
2000 ÷ 5000 = 40%
A: The price has been reduced by 40%.
Example 5, (Angle of view of test point)
A project was originally planned to be completed in 10, but the task was actually completed in 8 days. What percentage is the actual daily repair more than originally planned?
The first network of new curriculum standards
Analysis and solution: According to "the original planned completion time is 10 day", we can get that: the original plan is to complete the project every day; According to the "8-day actual completion", you can get: actually complete the project every day. By using "the quantity actually completed per day is greater than the originally planned quantity ÷ the originally planned quantity per day", we can find out the percentage of actual maintenance per day.
( - ) ÷ = 25%
A: Actually, the daily maintenance volume is 25% more than originally planned.
Comments: Finding the correct quantitative relationship is the key to solve this problem. What is required in the question is the amount of tasks completed every day, not 10 and 8, because 10 and 8 are working hours, and it is easy to make mistakes when answering.
Example 6 (Calculation Method of Taxable Amount)
The total business of Yimin Hardware Company last year was 4 million yuan. If the business tax is paid at 3% of the turnover, how much business tax should be paid last year?
Analysis and solution: If the business tax is paid at 3% of the turnover, the turnover shall be "1". Proportion of business tax paid to turnover
3%, which is 3% of 4 million yuan. Find the percentage of a number, and also use multiplication to calculate it. When calculating, the percentage can be calculated by part number or decimal.
400×3% = 400× 12 (ten thousand yuan)
Or 400× 3% = 400× 0.03 = 12 (ten thousand yuan)
A: Last year, the business tax was 6.5438+0.2 million yuan.
Comments: In the real society, various tax rates are different. The calculation of tax payable is basically to find out the percentage of a number.
Example 7 (Simple practical problems related to tax payable)
Uncle Wang bought a motorcycle worth 16000 yuan. According to the regulations, you have to pay 10% vehicle purchase tax when buying motorcycles. How much did Uncle Wang spend on this motorcycle?
Analysis and solution: The money uncle Wang needs to buy this motorcycle should include the purchase price and the vehicle purchase tax 10%, which accounts for 10% of the motorcycle purchase price. You can calculate the vehicle purchase tax to be paid first. You can also think of it this way: the vehicle purchase tax accounts for 10% of the purchase price, and the purchase price is regarded as "1". The money that Uncle Wang needs to buy this motorcycle is equivalent to the purchase price (1+ 16000 yuan), that is, 165438.
Methods1:16000×10%+16000 =1600+16000 =17600 (yuan).
Method 2:16000× (1+10%) =16000×1=17600 (yuan).
A: Uncle Wang will spend 17600 yuan on this motorcycle.
Example 8: A scenic spot in Yangzhou received 90,000 tourists during the "Eleventh" Golden Week in 2007, and the ticket revenue reached 270%.
Ten thousand yuan. According to 5% of the tickets, the business tax during the "Eleventh" Golden Week should be 0.45 million yuan.
Analysis and solution: Business tax is paid at 5% of the tickets, accounting for 5% of the ticket revenue, not 5% of the number of tourists.
A: Business tax should be paid 1.35 million yuan during the "Eleventh" Golden Week.
Example 1 (pre-tax interest settlement) Li Ming deposited 500 yuan money in the bank at one time for three years. How much interest should he get when it expires?
Deposit period (lump sum deposit and withdrawal) annual interest rate
3.87% a year
2 years 4.50%
5.22% in three years
Analysis and solution: According to the savings annual interest rate table, the three-year fixed annual interest rate is 5.22%.
Pre-tax interest = principal × interest rate× time
500× 5.22 %× 3 = 78.3 (Yuan)
A: The interest after maturity is 78.3 yuan.
Example 2 (After-tax Interest Settlement)
According to the national tax law, the interest earned by individuals in bank deposits should be taxed at the rate of 5%. Example 1 What is Li Ming's actual interest after tax?
Analysis and solution: After deducting the interest tax from the due interest, the rest is the earned interest.
Interest earned after tax = principal × interest rate × time × (1-5%)
500× 5.22 %× 3 = 78.3 yuan ... interest due.
78.3× 5% = 3.9 15 (yuan) ... interest tax
78.3–3.915 = 74.385 ≈ 74.39 (yuan) .................................................................................................................................
Or 500× 5.22 %× 3× (1-5%) = 74.385 (yuan) ≈ 74.39 (yuan).
A: After paying taxes, Li Ming earned interest of 74.39 yuan.
Example 3: Fang Ming deposited RMB 1.500 yuan in the bank for two years with an annual interest rate of 4.50%. After two years, you have to pay 5% interest tax to withdraw money. How much interest will you get after maturity?
Wrong answer:1500× 4.50 %× (1-5%) = 64.125 (yuan) ≈ 64. 13 (yuan).
Analysis of reasons: interest earned after tax = principal × interest rate × time × (1-5%), in which time was missed.
Correct answer:1500× 2× 4.50 %× (1-5%) =128.25 (yuan)
A: After maturity, the net interest is 128.25 yuan.
Comments: Interest tax is sometimes deducted according to the actual situation. According to national regulations, the interest tax rate is 5%, so interest is divided into pre-tax interest and after-tax interest. Pay attention to the distinction when doing the problem. But some of them do not need to pay interest tax, such as national construction bonds and education savings.
Example 4: The current price of a book is 6.4 yuan, which is cheaper than the original price 1.6 yuan. How much is the discount on this book?
Analysis and solution: How much discount to give To know how much the actual selling price is the original price, just divide the actual selling price by the original price.
6.4+ 1.6 = 8 (yuan)
6.4 8 = 20% off = 20% off
This book is sold at a 20% discount.
Comments: Ten percent discount, ten percent discount. The lower the discount of the same product, the lower the price. In the subject of discount, several discounts are sold at dozens of discounts from the original price, which does not mean increase or decrease.
Example 5. (Find the original price of a known discount)
"National Day" shopping mall promotion, a suit is sold at a 15% discount of 1020 yuan. What is the original price of this suit?
Analysis solution: 15% discount, that is, the actual selling price is equivalent to 85% of the original price. 85% of the known original price is 1020 yuan. What is the original price? You can work out an equation to solve it.
Original price × 85% = actual selling price
Solution: the original price of this suit is X yuan.
x × 85% = 1020
x = 1020 ÷ 85%
x = 1200
Inspection: (1) Divide the current price by the original price to see if there is a 15% discount.
1020 ÷ 1200 = 0.85 = 85%
(2) See if 85% of the original price is 1020 yuan.
1200× 85% = 1020 (yuan)
After testing, the answer conforms to the meaning of the question.
A: The original price of this suit is 1200 yuan.
An LCD TV is 6000 yuan. If you give a 15% discount, you can reduce the price by 2000 yuan.
Reason for analysis: 6000 yuan is the original price. If you give a 15% discount, you should first calculate the actual selling price and then subtract it, or calculate the price reduction first, accounting for 25% of the original price.
Correct answer: 6000-6000× 75% = 1500 (yuan)
Or 6000× (1-75%) =1500 (yuan)
A: You can reduce the price 1500 yuan.
Example 7 (Simple practical problems related to tax payable)
A batch of refrigerators, originally priced at 2000 yuan, are now sold at a 10% discount. When the customer buys them, he asks for a discount of 10%. How much will it cost if the transaction can be concluded?
Analysis and solution: "10% discount on promotion" means selling at 90% of the original price, using "original price × 90%", and "10% discount again" means 10% discount on the promotion price, and the promotion price should be multiplied by 90%.
2000× 90% × 90%
= 1800× 90%
= 1620 (yuan)
A: If business can be concluded, the price will be 1620 yuan.
Comments: The key point of the topic is that "another 10% discount" means another 10% discount on the promotion price, and the unit "1" is the promotion price, that is, the price after 10% discount on the original price. This is an error-prone point, so pay more attention to it.
Example 8, (Angle of View of Test Point)
The store lost 20% when it sold a commodity at the price of 40 yuan. What is the original price of this commodity and the loss?
Analysis and solution: selling at the price of 40 yuan means that the actual selling price is 40 yuan; Lost 20%, that is, the original price lost 20%, so the actual selling price is equivalent to the original price (1-20%).
Solution: Let the original price of this commodity be X yuan.
x × ( 1 - 20%) = 40
x × 80% = 40
x = 50
50× 20% = 10 (yuan)
A: The original price of this commodity is 50 yuan, 10 yuan.
Example 9, (Angle of View of Test Point)
A store sells two goods at the same time, each 30 yuan, one of which gains 20% and the other loses 20%. Does this store usually sell these two commodities at a profit or a loss? how much is it?
Analysis and solution: the profit is 20%, that is, the selling price is the cost price (1+20%); Loss of 20%, that is, the selling price is the cost price (1-20%). The prices of the two commodities are both within 30 yuan, so the cost prices of the two commodities can be calculated separately.
30 ÷ (1+20%) = 25 (yuan)
30 ÷ (1-20%) = 37.5 yuan.
25+37.5 = 62.5 (Yuan)
62.5–60 = 2.5 (Yuan)
Answer: The store sold these two commodities at a loss, and lost 2.5 yuan.
Example 1, (solving the sum and multiplication problem of sequence equations)
A rope is 48 meters long and cut into two sections, A and B, in which the length of rope B is 60% of that of rope A.. How long are ropes A and B?
Analysis and solution: the length of rope B is 60% of that of rope A, and the length of rope A is regarded as "1".
X meter
Armored rope
?
() meters? 48 meters
B rope
Rope b is 60% of rope a.
Equivalent relationship: length of rope A+length of rope B = total length.
Answer: If the length of rope A is x meters, then the length of rope B is 60% x meters.
x + 60%x = 48
1.6x = 48
x = 30
60%x = 30 × 60% = 18
Answer: The length of rope A is 30m, and the length of rope B is18m.
Inspection: 30+ 18 = 48 (meters), which is in line with the length of 48 meters for rope A and rope B.
18 ÷ 30 = 60%, and the length of rope b is 60% of rope a. ..
Example 2, (Solving Differential Time Problem with Column Equation)
The number of volleyball in the gymnasium is 75% of that in basketball, and basketball is 6 more than volleyball. How many basketball and volleyball are there?
Analysis and solution: The number of volleyball is 75% of basketball, and the number of basketball is regarded as the unit "1".
X piece
basketball
() pieces? There are six more
volleyball
The number of volleyball is 75% of that of basketball.
Equivalence: Basketball-Volleyball = 6.
Answer: If there are X basketballs, there are 75% X volleyballs.
x - 75%x = 6
0.25x = 6
x = 24
75%x = 24 × 0.75 = 18
A: Basketball 24, volleyball 18.
Will you test it yourself?
Test: 24- 18 = 6 (pieces), which means there are 6 more basketball than volleyball.
18 ÷ 24 = 75%, and the number of volleyball matches is 75% of that of basketball.
Comments: When solving the problems of sum times and difference times with the column equation method, we should pay attention to finding the quantity of the unit "1". Usually, the quantity of the unit "1" is set as x, and then another quantity is represented by the relationship between another quantity and the unit "1". Finally, the equations are listed according to their sum or difference.
Example 3: There are 40 fewer boys than girls in the sixth grade, and the number of girls in the sixth grade is equivalent to 140% of the number of boys. How many boys are there in the sixth grade?
Wrong solution: suppose there are x girls, 140% x boys.
140%x - x = 40
0.4x = 40
x = 100
140% x = 100× 1.4 = 140
Analysis and solution: According to "the number of girls in grade six is equal to 140% of the number of boys", the number of boys can be regarded as "1" unit. Let the number of boys be x and the number of girls be 140%. According to "the number of boys in grade six is 40 less than that of girls", a quantitative relationship can be obtained: "the number of girls"
Correct answer: If there are x boys, there are 140% X girls.
140%x - x = 40
0.4x = 40
x = 100
A: Boys are 65,438+000.
Comments: The reason for solving this problem is that the unit "1" is wrong. Remember that when looking for the unit "1", you should first look for the fraction (percentage), because without the fraction, there is no unit "1" and you can't see the "ratio", and the quantity after the "ratio" is.
Example 4, (column equation solves "what is a number that is known to be a few percent smaller than a number?" Find this number ")
There are 36 white rabbits, 20% less than gray rabbits. How many gray rabbits are there?
Analysis and solution: White rabbits are 20% less than grey rabbits, and grey rabbits are regarded as "1".
only
Grey rabbit
36?
white rabbit
20% less than the gray rabbit.
Equivalence relation: the number of grey rabbits-the number of small white rabbits is less than the number of grey rabbits = the number of small white rabbits.
Answer: suppose there are x gray rabbits.
x - 20%x = 36
0.8x = 36
x = 45
A: There are 45 gray rabbits.
Test: 45–45× 20% = 36 or (45–36) ÷ 45 = 20%, which meets the meaning of the question.
Example 5, (The equation of series solves "What is a number that is known to be more than a number?"? Find this number ")
There are 48 white rabbits, 20% more than gray rabbits. How many gray rabbits are there?
Analysis and solution: white rabbits are 20% more than gray rabbits, and gray rabbits are regarded as "1".
only
Grey rabbit
20% more than the gray rabbit.
?
white rabbit
48
Equivalence relation: number of grey rabbits+number of white rabbits more than grey rabbits = number of white rabbits.
Answer: suppose there are x gray rabbits.
x + 20%x = 48
1.2x = 48
x = 40
A: There are 40 gray rabbits.
Test: 40+40× 20% = 48 or (48–40) ÷ 40 = 20%, which meets the meaning of the question.
Comments: As in the previous example, we all seek the unit "1". When solving a problem, we should also pay attention to the exact number of units "1", depending on what the problem is seeking, and determine the calculation method.
Example 6 (Breakthrough of Difficulties)
If a commodity is sold at the current price 18 yuan, it will lose 25%. What is the original cost? If you want to make a profit of 25%, at what price should you sell the goods?
Analysis and solution: whether it is a loss of 25% or a profit of 25%, the unit "1" is the cost of the commodity. So we must first find the cost of this commodity. 18 yuan loses 25%, which means that 18 yuan is 25% less than the cost, which is the cost (1-25%). The profit is 25%, which means that the profit is 25% of the original cost, and the actual selling price is (1+25%).
Answer: Let the original cost be X yuan.
x - 25%x = 18
0.75x = 18
x = 24
24× (1+25%) = 30 (yuan)
Answer: The original price is 24 yuan, so the goods should be sold in 30 yuan.
Comments: In general, the profit and loss of goods is "1" in cost. The key to solve this problem is to determine the unit "1", which is also the most important when solving percentage application problems.
Example 7, (Angle of View of Test Point)
The fruit wholesale department will bring in a batch of fruits, accounting for 22% of the total for the first time, 0/.5 tons for the second time and 62% for the second time. How many tons of fruit are there?
Analysis and solution: According to the meaning of the question, you can draw the following line graph:
62%
22% for the first time 1.5 tons
" 1"? many
As can be seen from the figure, the tonnage of the two shipments-the tonnage of the first shipment = 1.5 tons, and the unit of "1" is the total tonnage of this batch of fruits. If this batch of fruit has x tons, then 62% x tons will be shipped twice and 22%x tons will be shipped for the first time.
Solution: suppose this batch of fruit has x tons.
62%x - 22%x = 1.5
40%x = 1.5
x = 3.75
This batch of fruit has 3.75 tons.