Question

Два землероби Іван і Григорій можуть зорати поле за 3 години. За скільки годин Григорій може зорати все поле, якщо Іван всю роботу може зробити на 8 годин швидше, ніж Григорій?

Answer 1

Приймемо за 1 все поле. Нехай Григорій може зорати все поле х годин, тоді Іван - (x+8) годин. Григорій і Іван зорали за $\bigg(\dfrac{1}{x} +\dfrac{1}{x+8} \bigg)$ годин, що за умовою задачі становить 1/3 години

$\dfrac{1}{x} +\dfrac{1}{x+8} = \dfrac{1}{3} ~~~|\cdot 3x(x+8)\\ \\ 3(x+8)+3x=x(x+8)\\ \\ 3x+24+3x=x^2+8x\\ \\ x^2+2x-24=0$

За теоремою Вієта:

          $x_1=-6$ не задовольняє умові

          $x_2=4$ години все поле Григорій може зорати.

Відповідь : за 4 години.

Answer 2

Problem Analysis

We are given that two farmers, Ivan and Grigory, can plow a field together in 3 hours. We need to determine how long it would take Grigory to plow the entire field if Ivan can do the entire job 8 hours faster than Grigory.

Solution

Let's assume that Grigory takes x hours to plow the entire field. Since Ivan can do the entire job 8 hours faster than Grigory, Ivan would take (x - 8) hours to plow the entire field.

We are given that when Ivan and Grigory work together, they can plow the field in 3 hours. This means that their combined work rate is 1/3 of the field per hour.

Using the concept of work rate, we can set up the following equation:

1/x + 1/(x - 8) = 1/3

To solve this equation, we can multiply both sides by 3x(x - 8) to eliminate the denominators:

3(x - 8) + 3x = x(x - 8)

Simplifying the equation:

3x - 24 + 3x = x^2 - 8x

6x - 24 = x^2 - 8x

Rearranging the equation:

x^2 - 14x + 24 = 0

Now we can solve this quadratic equation to find the value of x.

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -14, and c = 24.

Plugging in the values:

x = (-(-14) ± √((-14)^2 - 4 * 1 * 24)) / (2 * 1)

Simplifying:

x = (14 ± √(196 - 96)) / 2

x = (14 ± √100) / 2

x = (14 ± 10) / 2

We have two possible solutions:

1. When x = (14 + 10) / 2 = 24 / 2 = 12 2. When x = (14 - 10) / 2 = 4 / 2 = 2

Since we are looking for the time it takes Grigory to plow the entire field, we discard the solution x = 12 because it would mean Ivan takes negative time to complete the job.

Therefore, Grigory takes 2 hours to plow the entire field.

Answer

Grigory can plow the entire field in 2 hours if Ivan can do the entire job 8 hours faster than Grigory.