Question

Если дневную норму первого рабочего увеличить на 40%, а второго-на 20%, то за один день они изготовят 159 деталей.Если дневную норму первого рабочего увеличить на 20%, а второго-на 40%, то за один день они изготовят 153 детали. Найдите дрневную норму каждого рабочего

Answer 1

Первоначально:
Норма I рабочего  -  х дет./сутки
Норма II рабочего  -   y  дет/сут
Первая часть задачи:
Увеличим норму  
I рабочий  (х+0,4х) = 1,4х  дет/сут.      (40%=40/100=0,4)
II рабочий  (у+0,2у) =1,2у  дет/сут.       (20%=20/100=0,2)
Уравнение   1,4х +1,2у = 159
Вторая часть задачи:
I рабочий   1.2x   дет./сут.
II рабочий  1.4y  дет./сут.
Второе уравнение:  1,2х +1,4у=153
Система уравнений:
{ 1.4 x + 1.2y = 159           | *5
{ 1.2x  + 1,4у = 153           | *5

{7x+6y= 795                     |*(-6)       ⇒ х=(795-6у)/7
{6x +7y = 765                   |*7

{-42x -  36y= -4770
{42x + 49y= 5355
Метод сложения:
-42х - 36у +42х+49у= -4770 + 5355
13у= 585
у=585/13
у= 45  (дет./сутки) норма II рабочего
х= (795-6*45)/7= (795-270)/7 = 525/7=75 (дет./сут.) норма I рабочего

Ответ: 75 деталей дневная норма первого рабочего, 45 деталей - второго.

Answer 2

Calculation of Daily Norm for Each Worker

Let's denote the daily norm for the first worker as x and the daily norm for the second worker as y.

According to the given information, if the daily norm for the first worker is increased by 40% and the daily norm for the second worker is increased by 20%, they will produce 159 parts in one day. This can be expressed as the following equation:

1.4x + 1.2y = 159 Similarly, if the daily norm for the first worker is increased by 20% and the daily norm for the second worker is increased by 40%, they will produce 153 parts in one day. This can be expressed as the following equation:

1.2x + 1.4y = 153 To find the daily norm for each worker, we can solve this system of equations.

Solving the System of Equations

We can use various methods to solve this system of equations, such as substitution or elimination. Let's use the elimination method to solve it.

Multiplying the first equation by 1.2 and the second equation by 1.4, we get:

1.68x + 1.44y = 190.8 (Equation A) 1.68x + 2.1y = 214.2 (Equation B)

Subtracting Equation A from Equation B, we eliminate the variable x:

(1.68x + 2.1y) - (1.68x + 1.44y) = 214.2 - 190.8

Simplifying the equation, we have:

0.66y = 23.4

Dividing both sides of the equation by 0.66, we find:

y = 35

Now, we can substitute the value of y back into one of the original equations to find the value of x. Let's use the first equation:

1.4x + 1.2(35) = 159

Simplifying the equation, we have:

1.4x + 42 = 159

Subtracting 42 from both sides of the equation, we get:

1.4x = 117

Dividing both sides of the equation by 1.4, we find:

x = 83.57

Therefore, the daily norm for the first worker is approximately 83.57 and the daily norm for the second worker is 35.

Please note that due to rounding, the values provided are approximate.