Mba-day20 indefinite equation problems - exercises

1. Train of thought

1） Set unknowns , Set up equations
2） There are more unknowns than equations
3） Parity and division multiples

2. Question type

2.1 Example 1

In the year-end love giving activities , Shared by a company 100 People participated in the donation , Statistics , The total contribution is 19000 element , Personal donations include 100 element ,500 Genna 2000 Three kinds of yuan , Ask the unit to donate 500 Human resources are （）
A 13
B 18
C 25
D 30
E 38

answer ：A

 Explain ：
 Set personal donations to have 100 element ,500 Genna 2000 The number of yuan is x,y,x
x+y+z = 100
100x+500y+2000z = 19000 -> x + 5y+ 20z = 190

 elimination x
4y + 19z = 90
 even numbers + even numbers  =  even numbers 
4y( even numbers * Even or odd , y Even or odd numbers )
19z( Odd number * even numbers ,z  It must be an even number )
 When z=2, 4y+38=90 -> 4y=52 y = 13
 When z=4, 4y+76=90 -> 4y=14,y  Not divisible 
 When y=6,  dissatisfaction 

Example 2 Indefinite equation problem ：144a + 15b = 120, seek a,b

 Explain ：
144a + 15b = 720 -> 48a + 5b = 260
 even numbers + even numbers  =  even numbers 
b = 2, 48a + 10 = 260, 48a = 250,  Not divisible 
b = 4, 48a + 20 = 260,  48a = 240,  Not divisible 
a = 5, b = 4

Example 3 Indefinite equation problem ：44a + 15b = 720

44a + 15b = 720
 even numbers + even numbers  =  even numbers ,  But every value is valued , Too much work 

 Observe ：15： Divisibility  3,15, 720  Divisibility  3, 15
 Make  a = 15, 660 + 15b = 720 -> b = 4  Satisfy even number 

Example 4

A worker will 99 Marbles are packed in two boxes , Each largest box contains 12 star , Each small box contains 5 star , It happens to be finished , The number of known boxes is greater than 10, Then the small box has （） individual .
A 10
B 11
C 12
D 13
E 15

answer ：15

 Explain ：
 Set a big box a individual , Small box b individual 
12a + 5b = 99
a + b > 10

 According to parity ：12a  Constant is even ,99 It's odd , so 5b It must be an odd number .
b：1,3,5,7,9,11,13,15

 When b=11 when ,12a+55=99 -> 12a=44 -> a  Non integer , dissatisfaction 
 When b=13 when ,12a+65=99 -> 12a=34 -> a  Non integer , dissatisfaction 
 When b=15 when ,12a+75=99 -> 12a=24 -> a = 2

 Big box 2 individual , Small box 15 individual 

Example 5

Some kind of exam has been held 24 Time , A total of 426 Problem , Each time the questions are 25 topic 、16 Question or 20 topic , Then the high school entrance exam 25 Yes （） Time .
A 1
B 2
C 3
D 4
E 5
answer ：B

 Explain ：
 Out 25 topic 、16 topic 、20 The times of the questions are a,b,c
a+b+c = 24
25a+16b+20c = 426

 According to parity ,16b Constant even number ,20c Constant even number ,426 For the even ,25a Must satisfy even numbers , so a Cannot be odd , Can only be even .

 so a May be 2,4
 When a = 2  The results are as follows 
b + c =22 -> 20b + 20c = 440
16b + 20c = 376
-> 4b = 64 -> b = 16, c = 8

 When a=4 when ：
b + c = 20 -> 20b + 20c = 440
16b+20c=326
->4b = 113 -> b  Non integer , dissatisfaction 

 so a by 2, Examination 25 Yes （2） Time 

Example 6

Yes A,B,C Three kinds of toys ,A toy 3 Yuan a ,B toy 2 Yuan a ,C toy 1 Yuan a , It is known that three kinds of toys cost a total of 100 element , There are at least three kinds of toys 50 individual .
1）A The number of toys is the least
2）B The number of toys is the largest
Ask for conditions 1,2 The sufficiency of

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

 Explain ：
3a + 2b + c = 100  To launch  a + b + c >= 50

1）A The number of toys is the least , That is, the smallest positive integer  a =1
2b + c = 97
 even numbers +c =  Odd number 
 namely c:1,3,5,7,9,10
 When c = 1, b = 48,  Satisfy a + b + c >= 50
 When c = 3, b = 47,  Satisfy  a + b + c >= 50
 When c = 95, b =1,  Satisfy  a + b + c >= 50
 I.e. conditions 1 Alone enough 

2）B The number of toys is the largest  ->  In extreme conditions ： Make a=c=1
3a + 2b + c = 100
3+ + 2b + 1 = 100
b = 48 ->  Satisfy a + b + c >= 50

 I.e. conditions 1 Alone enough 

Example 7

Pony uses takeout software to buy barbecue , Fish tofu is known 2 A string of yuan , Mutton string 4 A string of yuan , Chicken wings 7 A string of yuan , He bought a total of 12 strand , The quantity of mutton kebabs can be determined
1） It took 42 element
2） bought 2 String chicken wings

answer ： Conditions 1 And conditions 2 Not enough alone , Conditions 1 and 2 Unite to fully

 Explain ： Set fish and tofu separately 、 Mutton string 、 The number of chicken wings is a, b, c
a+b+c = 12

1)  It took 42 element 
2a + 4b + 7c = 42 ->  even numbers + even numbers +7c =  even numbers  -> 7c  It must be an even number ,c  It must be an even number , The number of mutton kebabs that cannot be rolled out alone 

2） bought 2 String chicken wings  -> a+b=10,  The quantity of mutton kebabs cannot be introduced separately 

1） and 2） Conditional Union 
a+b=10
2a + 4b + 7c = 42 ->2a + 4b = 28 -> a + 2b = 14

b = 4, a = 6

 Therefore, the mutton kebab can be determined as 4 strand .

 I.e. conditions 1 And conditions 2 Not enough alone , Conditions 1 and 2 Unite to fully 

Example 8

A school held a rope skipping competition , Planners plan to use the most 50 Yuan to buy A,B Two kinds of skipping rope , among A The unit price 6 element , Plan to buy at least 3 root ,B The unit price 7 element , Plan to buy at least 2 root , There are different ways to buy （） Kind of
A 5
B 6
C 7
D 8
E 9

answer :C

 Explain ： Respectively A,B The purchase quantity of skipping rope is a,b
6a+7b <= 50 (a>=3, b>=2)

a=3, b=2,3,4 namely 3 Kind of 
a=4, b=2,3 namely 2 Kind of 
a=5, b=2, namely 1 Kind of 
a=6, b=2, namely 1 Kind of 

 To sum up the above ： common 7 Kind of 

Example 9

A factory produced 100 New parts , existing A,B,C Three types of machine tools need to be equipped with this component , a A need 12 individual ,B need 8 individual ,C need 5 individual , There are three known machine tools 14 platform , seek A Machine tool （） platform
A 3 platform
B 4 platform
C 6 platform
D 8 platform
E 10 platform

answer ：A

 Explain ：
a + b + c = 14 -> 5a + 5b + 5c = 70
12a + 8b + 5c = 100

-> 7a + 3b = 30 ->  Odd number + Odd number = even numbers , It's not very intuitive 


a + b + c = 14 -> 8a + 8b + 8c = 112
12a + 8b + 5c = 100

-> -4a + 3c = 12 -> - even numbers  + 3c =  even numbers  -> 3c  To be an even number , be c To be an even number 
 namely c=2,4,6,8,10
 When c = 2 -> -4a + 6 = 12 -> -4a = 6,a<0 dissatisfaction 
 When c = 4 -> -4a + 12 =12 -> a = 0  dissatisfaction 
 When c = 6 -> -4a + 18 = 12 -> -4a = -6, a  Non integer , dissatisfaction 
 When c = 8 -> -4a + 24 = 12 -> -4a = -12, a = 3,  Satisfy 
 When c = 10 -> -4a + 30 = 12 ....

Example 10

A department organizes employees to participate in tree planting ,10 People plant trees together 93 star , Everyone is known to have at least one species 8 A tree , At most 10 A tree , Kind of 10 The number of employees of the tree is higher than that of the tree 9 The tree has the largest number of employees （） people .
A 2
B 3
C 4
D 5
E 6

answer ：D

 Explain ：
 Set up species respectively 8、9、10 The man of the tree a,b,c
a + b + c = 10 -> 8a + 8b + 8c = 80
8a + 9b + 10c = 93
-> b + 2c = 13 -> b +  even numbers  =  Odd number , namely b It must be odd ,b  by  1,3,5,7,9
 When b = 1, c = 6,  Kind of 10 The number of employees of the tree is higher than that of the tree 9 The tree has a large number of employees 5 people ,  The answer to this question , At most 5 people 
 When b = 3, c = 5,   Kind of 10 The number of employees of the tree is higher than that of the tree 9 The tree has a large number of employees 2 people 
 When b = 5. c = 4

Example 11

The bread sold in the West Point shop is 5 A bag or 3 A bag of , Do not open retail , It is known that 5 A bag of 8 element ,3 The price of a bag is 5 element , The minimum amount of money used is 76 element .
1） To give 47 Students each send 1 A loaf of bread
2） To give 50 Students each send 1 A loaf of bread
Find the sufficiency

answer ： Conditions 1 to the full , Conditions 2 inadequate

 Explain ：
 According to the meaning ： The principle of doing more with the least money , Just buy more bread , Buy more 5 A bag of 8 Yuan of bread 

1） To give 47 Students each send 1 A loaf of bread , And try to buy as many bags as possible 5 A loaf of bread is the cheapest 
5x + 3y = 47
 Get by exhausting ：x = 7, y= 4  The minimum amount of money used to meet the requirements is 76 element 

2） To give 50 Students each send 1 A loaf of bread , And try to buy as many bags as possible 5 A loaf of bread is the cheapest 
 If the most economical way , Buy it all 5 A bag of 8 element ,50 One for each student , At least it takes 80 element , The minimum amount of money that cannot be rolled out is 76 element 

Example 12（ Unclear sum of digits ）

A box of books , To give evenly to 6 Two children , More than 1 Ben . To give evenly to 8 Two children , More than one , To give evenly to 9 Two children , More than 1 Ben , This box of books is the least m Ben , be m The sum of the digits of is （）
A 10
B 3
C 4
D 5
E 9

answer ：A

 The sum of numbers ： Add up the numbers on each digit , The result is the sum of the numbers on each digit . for example ,123456 The sum of the numbers on each digit of is 1+2+3+4+5+6=21.

 Explain ： Set equal points respectively 6 Share ,8 Share ,9 Each child gets a Book of a, b, c
6a + 1 = m ->  even numbers  + 1 ->  Odd number 
8b + 1 = m ->  even numbers  + 1 ->  Odd number 
9c + 1 = m -> 9c + 1 ->  Odd number , c  It must be an even number 
 Make c = 2,  be  m=19,  The sum of numbers ：1+9=10
 Make c = 4,  be  m=37,  The sum of numbers ：2+8=10
 Make c = 6,  be  m=55,  The sum of numbers ：2+8=10

 answer ：A