Mba-day23 at most at least questions - exercises

1. Train of thought

When analyzing an object, at least （ at most ） when , It can be converted into the rest at most （ least ） To analyze

• 1 Each object is at most
• 2 Let the whole at most be at least
• 3 At most, the individual should at least
  

2. Example

2.1 Example 1（ Wrong questions ）

Shared by a class 40 people , Among them, I like playing table tennis 、 Basketball 、 The students of volleyball are 35, 33, 32 people , Ask students who like all three sports to have at least （） people

answer ： 20

 One of them likes playing table tennis 、 Basketball 、 The students of volleyball are  35, 33, 32  people 
 have to ： I don't like playing table tennis 、 Basketball 、 The students of volleyball are  5, 7, 8  people 
-> The number of people who dislike at least one sport is at most ：5+7+8 =20 
-> Students who like all three sports have at least  40 - 20 = 20

======================================================
 Explain ：（ It didn't work out ）
 Like table tennis = Only like table tennis + I like table tennis and basketball + I like table tennis and volleyball  +  Like table tennis 、 Basketball and volleyball  = 35
 Like basketball = I only like basketball + I like table tennis and basketball + I like basketball and volleyball  +  Like table tennis 、 Basketball and volleyball  = 33
 I like volleyball = I only like volleyball + I like volleyball and basketball + I like table tennis and volleyball  +  Like table tennis 、 Basketball and volleyball  = 32

 namely   like 1 Kind of  +  like 2 Kind of  +  like 3 Kind of , If you want to like 3 The least species , Like 1 Species and 2 Number of species , Do not like should be the most .
3* like 1 Kind of  + 6* like 2 Kind of  + 3 *  like 3 Kind of  = 35 + 33 + 32 = 100  people 

 I like 1 Kind of 、2 Kind of 、3 Number of species , They don't like to be a, b, c, d -> c  At least , a + b + d  at most 
3a + 6b + 3c = 100
a + b + c + d = 40 -> 3a + 3b + 3c + 3d = 120 -> 3a + 6b + 3c - 3b + 3d = 120
->
100  - 3b + 3d = 120 -> 3d - 3b = 20 -> 3(d-b) = 20  because bd Integers , Therefore, this obviously does not hold 
bd: Must be both even and odd numbers 
1,3,5,7,9,11,13,15,17,19,21
2,4,6,8,10,12,14,16,18,20

2.2 Example 2（ Wrong questions ）

share 100 People take the recruitment examination of a company , The examination contents are 5 Problem .1 to 5 The questions are 80, 92, 86, 78, 74 Answer , bingo 3 Problem sum 3 People with more than questions can pass the exam , So at least （） One can pass the exam .

answer ： At least （70） One can pass the exam

 Refer to the answer ideas 
 Explain ：
 All right ：80+92+86+78+74 = 410
 If all is right ：500  topic 
->  So there are  500 - 410 = 90  The question is not done correctly 

 If the number of people who fail to pass is the largest , That is everyone's fault 3 Avenue , at most  90/3 = 30  People fail .
->  At least  100 - 30 = 70  People pass through 


 Wrong thinking 
 Explain ：
1 to 5 The questions are 80, 92, 86, 78, 74  Answer 
1 to 5 The questions are 20, 08, 14, 22, 26  Not right 
 at most 4,5 The most wrong answers ：22, 26, Remove duplicate personnel , That is, the biggest 26 people , At least （100-26=74） One can pass the exam 

2.3 Example 3

5 Contestants won a total of... In a math contest 404 branch , The scores are not equal to each other , And the player with the highest score scores 90 branch , Then the player with the lowest score scores at least （） branch

answer ： 50

 player ： nail  <  B （87） <  C （88） <  Ding （89）<  E  （90）
 The player with the lowest score scores at least （404-90-89-88-87 = 50） branch 

2.4 Example 4

A knowledge contest has 50 Problem , Each correct answer will result in 3 branch , Wrong answer or no answer 1 branch , If a contestant wants at least 90 branch , So how many questions does he need to answer correctly ？

answer ： He must at least answer correctly 35 topic

 Explain ：
 Suppose the answer is correct x Avenue , answer incorrectly 50-x
3x + (50-x)*-1 >= 90
3x - 50 + x >= 90
4x >= 140
x >= 35

2.5 Example 5（ Wrong questions ）

5 The average score of students who took the 100 point test was 91 branch , Their grades are not equal to each other . Then the third place according to the score from high to low is at least 89 branch
1） The highest 96 branch
2） Lowest score 87 branch

answer ： Conditions 1 to the full , Conditions 2 to the full

 Explain ：
1） The highest  96  branch 
5  The average score of students who took the 100 point test was 91 branch , ->  Total score ：455
 nail  <  B  <  C  <  Ding <  E  （96）, The lowest score for nail pushing 
 If you want to be third, at least , Others should be as tall as possible , Namely Ding =95, Set the third place and x
 nail (x-2) <  B (x-1)  <  C (x) <  Ding （95）<  E  （96）
-> x = 89  I.e. conditions 1 to the full 
 nail (87) <  B (88)  <  C (89) <  Ding （95）<  E  （96）

2） Lowest score  87  branch 
 Same condition 1, That is, the condition is sufficient