# Work to time ratio! [closed]

Once Valera left the house, walked to the villa, painted 11 fence boards there, and returned home 2 hours after leaving. Another day, Valera went to the villa with Olga, together they painted 8 fence boards (without helping or interfering with each other), left together and returned home 3 hours after leaving it. How many boards can Olga paint alone if she needs to return home an hour and a half after leaving? The physical abilities of Valera and Olga, their hard work, and working conditions are unchanged

## 1 Answer

The only way to make sense of the numbers is to assume that Olga moves much slower than Valera. Let us write tV and tO for the time Valera or Olga needs to get to the villa and back. And let us write vV and vO for their work rates, fence boards per hour. We then get two formulas:
(1) (2h - tV) x vV = 11
(2) (3h - max(tO, tV) ) x (vO + vV) = 8

Because the unknown times and rates are nonnegative this yields inequalities:
(1') 2h x vV >= 11
(2') (3h - tO) x vV <= 8

It follows:
(3h - tO)/2h <= 8/11
tO >= 2h x (3/2 - 8/11) = 17/11 h > 3/2 h

The answer is therefore

0. Olga is too slow to get there and back in the allotted time, hence she cannot get any work done.

• Sorry I did not understand your solution. What is $3h - t_O$ ? What is $T_0$ ? How did you conclude that $0$ is the answer? Why $2h \times V_v >= 11$ ? – MathR Apr 4 at 18:26
• @MathR as it says in the answer t_O is the time Olga needs to get to travel to the villa and back. Since Olga is the slower of the two it is also the time Olga and Valera need when they travel together. 3h-t_O is the time they have left for working. The inequalities are obtained from the equations by leaving out terms we know are nonnegative. – loopy walt Apr 4 at 19:03
• 2 x (3/2 - 8/11) = 2 x (33/22 - 16/22) = 2 x (17/22) = 17/11 – loopy walt Apr 4 at 19:10