Question

Мастер и его ученик должны были выполнить работу к определенному сроку. Однако когда была выполнена половина работы, ученик заболел, и мастер, оставшись один. закончил работу с опозданием на 2 дня. за сколько дней мог бы выполнить всю работу каждый из них, если мастеру потребовалось бы на 5 дней меньше, чем ученику?Методическое указание:Обозначить как переменный объем работы и производительности мастера и ученика.

Answer 1

Пусть х и у производительность мастера и ученика тогда  1/x+5=1/y
1/(x+y) - время по сроку.
1/2(x+y) -время половины работы
1/2x-время половины работы работы
1/2x-1/2(x+y)=2  1/x-1/(x+y)=4
y=(5x+1)/x
1/x-1/(x+x/5x+1))=4
1/5x^2+2x=4
20x^2+8x-1=0
x=(-4+6)/20=1/10
мастер выполнитработу за 10 дней, а ученик за 15

Answer 2

Problem Analysis

In this problem, we are given that a master and his apprentice were supposed to complete a task within a certain deadline. However, when half of the work was completed, the apprentice fell ill, and the master had to finish the task alone, resulting in a delay of 2 days. We are asked to determine how many days it would take for each of them to complete the entire task if the master required 5 days less than the apprentice.

To solve this problem, we can assign variables to represent the total amount of work and the productivity of the master and the apprentice. By setting up equations based on the given information, we can solve for the number of days it would take for each of them to complete the entire task.

Solution

Let's denote the total amount of work as W, the productivity of the master as M, and the productivity of the apprentice as A.

From the given information, we know that when the apprentice fell ill, the master completed half of the work and then finished the remaining half with a delay of 2 days. This can be represented by the equation:

0.5W = 0.5M + 2 We are also given that the master required 5 days less than the apprentice to complete the entire task. This can be represented by the equation:

W/M = (W/5) / A To solve these equations, we can substitute the value of W from equation into equation:

(0.5M + 2)/M = ((0.5M + 2)/5) / A

Simplifying this equation, we get:

10(0.5M + 2) = M(0.5M + 2)

Expanding and rearranging the equation, we get:

5M + 20 = 0.5M^2 + 2M

Simplifying further, we get:

0.5M^2 + 2.5M - 20 = 0

Now, we can solve this quadratic equation to find the value of M (productivity of the master).