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. ..
The number of volleyball in the gymnasium is 75% of the number of 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. (The 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 grey 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 solves "What is a number that is known to be a few percent more than a 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 of 18 yuan, the loss will be 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. From the perspective of test sites, the fruit wholesale department will bring in a batch of fruits, accounting for 22% of the total for the first time, 1.5 tons for the second time and 62% for the second time. How many tons are there in this batch of fruit?
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.
Comments: When solving slightly complicated percentage application problems, you should learn to draw line drawings. Its advantages are: making the conditions of the problem simple and finding the quantitative relationship easier and more convenient. When drawing, you should first find out the quantity of the unit "1", then use the line segment to represent the quantity of the unit "1", and then represent other quantities.
First, basic training:
1. Find the unit "1" in the following question.
① The number of boys accounts for 60% of the number of girls.
② The number of boys is 20% more than that of girls.
③ The number of girls is 25% less than that of boys.
(4) Processing a batch of parts, 80% of which has been completed.
The unit price of pork this year has increased by 80% compared with last year.
2. According to the given information, tell the equal relationship between quantity and quantity.
(1) 60% of the total length of a road has been repaired.
② The current price of a color TV is lower than the original price 10%.
③ There are more pine trees 13 than cypress trees.
3. Look at the chart.
With 30%? only
Grey rabbits are 25% more than grey rabbits.
Use it? There are 28 tons of white rabbits left.
30
4, column calculation:
(1) 75% of a number is more than 25% of 30 1.5. Find this number.
② 25% of a number is 30 and less than 75% of it. Find this number.
Second, solve the problem:
1, contrast exercise
(1) A factory used 60 tons of coal in June, 25% less than that in May. How many tons of coal were used in May?
(2) A factory used 60 tons of coal in June, and in May it used 25% more coal than in June. How many tons of coal were used in May?
2. Desk is more expensive than chair 10 yuan. If the unit price of the chair is 60% of the unit price of the desk, what are the unit prices of the desk and the chair respectively?
There are 360 pear trees and apple trees in the orchard, of which 20% are apple trees. How many apples and pears are there?
The price of a set of tables and chairs is 78 yuan, in which the price of chairs is 30% of that of tables. How much are these tables and chairs?
5, a rope, the first cut off 25% of the total length, the second cut off 35% of the total length, twice cut off 6 meters, how many meters is this rope?
6, a rope, the first cut off 25% of the total length, the second cut off 35% of the total length, the second cut off more than the first 1 meter, how long is this rope?
7. According to the question.
Pingshan Tea Farm originally planned to plant 20 hectares of tea last year, but actually planted 25 hectares. _ _ _ _ _ _?
(1) What is the proportion of hectares actually planted with tea?
(2) What is the proportion of hectares planned to grow tea?
(3) How many hectares is the actual tea planting area more than originally planned?
(4) What percentage of tea is planned to be planted less than that actually planted?
8. Fill in the conditions according to the formula.
There are 200 apple trees in the orchard. How many pear trees are there?
①200÷20%
②200×20%
③200÷( 1+20%)
④200÷( 1-20%)
⑤200×( 1-20%)
⑥200×( 1+20%)
Test answer
First, basic training:
1. Find the unit "1" in the following question.
① The number of boys accounts for 60% of the number of girls. Think of the number of girls as the unit "1"
② The number of boys is 20% more than that of girls. Think of the number of girls as the unit "1"
③ The number of girls is 25% less than that of boys. In terms of the number of boys, "1"
(4) Processing a batch of parts, 80% of which has been completed. "1" in a batch of parts.
The unit price of pork this year has increased by 80% compared with last year. Think of the unit price of pork last year as "1"
2. According to the given information, tell the equal relationship between quantity and quantity.
(1) A road, 60% × 60% of the total length has been repaired = repaired.
(2) a color TV, the current price is lower than the original price 10%, the original price × 10% = price reduction.
Original price ×( 1- 10%)= current price.
③ The number of pine trees is 13 cypress×13 = the number of pine trees is more than cypress trees.
Cypress× (1+13) = pine.
3. Look at the chart.
With 30%? only
Grey rabbits are 25% more than grey rabbits.
Use it? There are 28 tons of white rabbits left.
28 ÷ (1-30%) × 30% =12 (ton) 30.
x + 25%x = 30
x = 24
4, column calculation:
(1) 75% of a number is more than 25% of 30 1.5. Find this number. 75% x–30×25% = 1.5
x = 12
② 25% of a number is 30 and less than 75% of it. Find this number. 75% x–25% x = 30
x = 60
Second, solve the problem:
1, contrast exercise
(1) A factory used 60 tons of coal in June, 25% less than that in May. How many tons of coal were used in May?
Solution: We assume that X tons of coal will be used in May. x–25% x = 60
x = 80
(2) A factory used 60 tons of coal in June, and in May it used 25% more coal than in June. How many tons of coal were used in May?
60+60 × 25% = 75 (ton)
2. Desk is more expensive than chair 10 yuan. If the unit price of the chair is 60% of the unit price of the desk, what are the unit prices of the desk and the chair respectively?
Solution: the unit price of the desk is X yuan, and the unit price of the chair is 60%x yuan.
x–60% x = 10
x = 25
25 × 60% = 15 (yuan) or 25–10 =15 (yuan)
Answer: The unit price of desks is 25 yuan, and the unit price of chairs is 15 yuan.
There are 360 pear trees and apple trees in the orchard, of which 20% are apple trees. How many apples and pears are there?
Solution: Suppose there are X trees in pear trees and 20%x trees in apple trees.
x + 20%x = 360
x = 300
300 × 20% = 60 (tree) or 360–300 = 60 (tree)
A: There are 300 pear trees and 60 apple trees.
The price of a set of tables and chairs is 78 yuan, in which the price of chairs is 30% of that of tables. How much are these tables and chairs?
Solution: the unit price of the desk is X yuan, and the unit price of the chair is 30%x yuan.
x + 30%x = 78
x = 60
60 × 30% = 18 (yuan) or 78–60 =18 (yuan)
Answer: The unit price of desks is 60 yuan, and the unit price of chairs is 18 yuan.
5, a rope, the first cut off 25% of the total length, the second cut off 35% of the total length, twice cut off 6 meters, how many meters is this rope?
Solution: Let this rope be x meters long.
25%x + 35%x = 6
x = 10
This rope * * * is 10 meter long.
6, a rope, the first cut off 25% of the total length, the second cut off 35% of the total length, the second cut off more than the first 1 meter, how long is this rope?
Solution: Let this rope be x meters long.
35%x - 25%x = 1
x = 10
This rope * * * is 10 meter long.
7. According to the question.
Pingshan Tea Farm originally planned to plant 20 hectares of tea last year, but actually planted 25 hectares. _ _ _ _ _ _?
(1) What is the proportion of hectares actually planted with tea? 25 ÷ 20 = 125%
(2) What is the proportion of hectares planned to grow tea? 20 ÷ 25 = 80%
(3) How many hectares is the actual tea planting area more than originally planned? (25 – 20) ÷ 20 = 25%
(4) What percentage of tea is planned to be planted less than that actually planted? (25 – 20) ÷ 25 = 20%
8. Fill in the conditions according to the formula.
There are 200 apple trees in the orchard. How many pear trees are there?
① 200% of apple trees are 20% of pear trees.
②200×20% pear trees are 20% apple trees.
③200 apple trees (1+20%) are 20% more than pear trees.
④200( 1-20%) apple trees are 20% less than pear trees.
⑤200×( 1-20%) pear trees are 20% less than apple trees.
⑥200×( 1+20%) pear trees are 20% more than apple trees.