Question

Олжас и Дима нарубят дрова на зиму за 10 дней,Дима и Антон -за 15 дней,а Олжас и Антон - за 18 дней. За сколько дней нарубят дрова Олжас ?Дима?Антон?

Answer 1

Пусть х-производительность Олжаса, у-производительность Димы, z-производительность Антона. Работу примем за 1. Составляем систему уравнений, исходя из условий совместной работы ребят: (х+у)*10=1 (у+z)*15=1 (x+z)*18=1   10x+10y=1 I*3 15y+15z=1 I*2 18x+18z=1   30x+30y=3 вычитаем из первого ур-я второе 30y+30z=2 18x+18z=1    30x-30z=1 I*3      18x+18z=1 I*5    90x-90z=3 90x+90z=5  вычитаем из второго уравнения первое   90z-(-90z)=2 180z=2 z=1/90   1) 18х+18*1/90=1 отсюда х=2/45 2) 15у+15*1/90=1 отсюда у=1/18  Т.к. работу мы приняли за 1, то Олжас нарубит дрова за1: 2/45=45:2=22,5 (дней), Дима нарубит дрова за 1:1/18=18 (дней), Антон нарубит дрова за 1:1/90=90 (дней) Ответ:Олжас за 22,5 дней, Дима за 18 дней, Антон за 90 дней

Answer 2

Problem Analysis

We are given that Olzhas and Dima can chop firewood for the winter in 10 days, Dima and Anton can do it in 15 days, and Olzhas and Anton can do it in 18 days. We need to determine how many days it will take for Olzhas, Dima, and Anton to chop the firewood individually.

Solution

Let's assume that the amount of firewood chopped is the same in each scenario. We can set up a system of equations to solve for the individual chopping rates of Olzhas, Dima, and Anton.

Let's denote the chopping rate of Olzhas as O, Dima as D, and Anton as A. We can use the formula: rate = work / time.

From the given information, we can set up the following equations:

1. Olzhas and Dima can chop firewood in 10 days: - O + D = 1/10 (Equation 1)

2. Dima and Anton can chop firewood in 15 days: - D + A = 1/15 (Equation 2)

3. Olzhas and Anton can chop firewood in 18 days: - O + A = 1/18 (Equation 3)

We can solve this system of equations to find the individual chopping rates.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution.

From Equation 1, we can express D in terms of O: - D = 1/10 - O

Substituting this value of D into Equation 2, we get: - (1/10 - O) + A = 1/15

Simplifying this equation, we get: - A = 1/15 - 1/10 + O - A = (2 - 3)/30 + O - A = -1/30 + O

Substituting the value of A into Equation 3, we get: - O + (-1/30 + O) = 1/18

Simplifying this equation, we get: - 2O - 1/30 = 1/18 - 2O = 1/18 + 1/30 - 2O = (5 + 3)/90 - 2O = 8/90 - O = 4/90 - O = 2/45

Now that we have the value of O, we can substitute it back into Equation 1 to find D: - D = 1/10 - O - D = 1/10 - 2/45 - D = 9/90 - 4/90 - D = 5/90 - D = 1/18

Similarly, we can substitute the value of O into Equation 3 to find A: - A = 1/18 - O - A = 1/18 - 2/45 - A = 5/90 - 4/90 - A = 1/90

Answer

Based on the calculations, we have determined the chopping rates of Olzhas, Dima, and Anton: - Olzhas can chop firewood in 2/45 of a day. - Dima can chop firewood in 1/18 of a day. - Anton can chop firewood in 1/90 of a day.

Since the chopping rate is the reciprocal of the time taken, we can calculate the number of days it will take for each person to chop the firewood: - Olzhas will take 45/2 days. - Dima will take 18 days. - Anton will take 90 days.

Therefore, it will take Olzhas 45/2 days, Dima 18 days, and Anton 90 days to chop the firewood individually.