"The big tank needs filling before the trays go in," Comet says, hose in hand. "Alone, it takes me 30 minutes."
"The second hose takes me 20," Wren says. "Together, how long? What do you notice if we think in tank-per-minute?"
"You fill 1/20 of the tank each minute. I fill 1/30. Together we fill 1/20 plus 1/30 each minute."
Nova projects the sum over the tank. "Would you like a hint? Call the time together t, and the whole tank 1."
"So 1/30 + 1/20 = 1/t," Comet says. "Common denominator 60. 2/60 + 3/60 is 5/60, which is 1/12."
"Twelve minutes together," Wren says. "A rational equation, built from a job. Let me start the stopwatch."
Twelve minutes later the tank brims, and Comet writes the time in the log with a flourish.
| Glass House problem | Equation | Solution | Makes sense |
|---|---|---|---|
| Comet fills the tank in 30 min, Wren in 20. Time together t. | 1/30 + 1/20 = 1/t | t = 12 | 12 minutes |
| Dilute the 20-liter, 10% tank to 4% with x liters of water. | (200)/(x + 20) = 4 | x = 30 | 30 liters |
| One hose fills a trough in 40 min, another in 60. Time together t. | 1/40 + 1/60 = 1/t | t = 24 | 24 minutes |
Notice that two hoses together are always faster than the faster hose alone. If your answer is not, check the setup.
For the first row: 1/30 + 1/20 = 1/12, so 1/t = 1/12 and t = 12.
| Statement | True or false? |
|---|---|
| A rational expression is a polynomial divided by a polynomial. | ? |
| To add rational expressions you need a common denominator. | ? |
| (x + 5)/(x + 2) can be written as 1 + 3/(x + 2). | ? |
| An extraneous solution makes a denominator zero and is thrown out. | ? |
| Two hoses together take longer than the slower hose alone. | ? |
Terrific week. You can build and solve a rational equation and spot a fake solution. Tomorrow is Garden Day.