Question

Володя может покрасить забор на 3 часа быстрее, чем Серёжа. Если они будут красить этот забор вместе, то за 4 часа покрасят 7/9. За какое время Володя покрасит этот забор один?

Answer 1

Пусть за $x$ часов Володя может покрасить весь забор один,  тогда

         за  $x+3$ часов Серёжа может покрасить весь забор один.

$\frac{1}{x}$  красит Володя за 1 час;

$\frac{1}{x+3}$  красит Серёжа за 1 час

Уравнение

$4*(\frac{1}{x} +\frac{1}{x+3}) =\frac{7}{9}$       (ОДЗ: $x>0$)

$(\frac{1}{x} +\frac{1}{x+3}) =\frac{7}{9*4}$

$\frac{x+3+x}{x(x+3)} =\frac{7}{36}$

$\frac{2x+3}{x^2+3x} =\frac{7}{36}$

$7*(x^2+3x)=36*(2x+3)$

$7x^{2} +21x=72x+108$

$7x^{2} -51x-108=0$

$D=2601-4*7*(-108)=5625=75^2$

$x_1=\frac{51-75}{2*7}=\frac{-24}{14}=-1,7$  не удовлетворяет ОДЗ.

$x_2=\frac{51+75}{2*7}=\frac{126}{14}=9>0$  удовлетворяет ОДЗ.

Ответ: за 9 часов Володя может покрасить весь забор один.

Answer 2

Solution:

To solve this problem, we can use the concept of work rates and combine the individual work rates of Volodya and Sergey to find the time it takes for Volodya to paint the fence alone.

Let's denote: - Volodya's work rate as V (in fences per hour) - Sergey's work rate as S (in fences per hour)

We are given that Volodya can paint the fence 3 hours faster than Sergey, so we have the equation: V = S + 3

When they work together, they can paint 7/9 of the fence in 4 hours. This can be expressed as: 4(V + S) = 7/9

Now, let's solve for V to find out how long it takes for Volodya to paint the fence alone.

First, we can express Sergey's work rate in terms of Volodya's work rate: S = V - 3

Substitute S = V - 3 into the equation 4(V + S) = 7/9: 4(V + (V - 3)) = 7/9

Simplify the equation: 4(2V - 3) = 7/9

Now, solve for V: 8V - 12 = 7/9 8V = 7/9 + 12 8V = 7/9 + 108/9 8V = 115/9

Finally, solve for V: V = (115/9) / 8 V = 115/72

Answer:

So, Volodya can paint the fence alone in 115/72 hours, which is approximately 1.597 hours.