Rate Problems: Plugging in Rate Problems - Work as an Invisible Variable

Tom harvests a corn field in three hours while Luke harvests the same field in five hours. If Tom starts harvesting at 6:00 and Luke joins him at 7:00, at what time will both of them finish harvesting the field?

Incorrect. [[Snippet]]

Incorrect. Notice that the work on the field did not start at the same time for Tom and for Luke.

(2) Next, Tom and Luke work together with a combined rate of $$5+3=8$$ acres per hour. | | Rate | Time | Work | |-----------|----------------------|------------------|--------------------| | Tom | $$\frac{15}{3} = 5$$ | $$3$$ | $$15$$ | | Luke | $$\frac{15}{5} = 3$$ | $$5$$ | $$15$$ | | Tom alone | $$5$$ | $$1$$ | $$5 \times 1 = 5$$ | | Together | $$5+3 = 8$$ | $$\color{red}{\frac{10}{8}}$$ | $$15-5 = 10$$ | In this row, the time is equal to >$$\displaystyle \frac{10}{8} = 1.25 \text{ hours} = 1 \text{ hour and } 15 \text{ minutes}$$ Therefore, the field will be harvested 1 hour 15 minutes after 7:00, the beginning of the second phase, which brings us to 8:15.

Incorrect. You may have made an error only in the final step of converting time units.

Correct. [[Snippet]] Calculate the rate of Tom and Luke, based on the __Plug In__ of 15 acres per field: | | Rate | Time | Work | |------|----------------------|-------|--------| | Tom | $$\color{red}{\frac{15}{3} = 5}$$ | $$3$$ | $$\color{green}{15}$$ | | Luke | $$\color{red}{\frac{15}{5} = 3}$$ | $$5$$ | $$\color{green}{15}$$ |

Now there are two phases: (1) First Tom works alone from 6:00 to 7:00: | | Rate | Time | Work | |-----------|----------------------|-------|--------------------| | Tom | $$\frac{15}{3} = 5$$ | $$3$$ | $$15$$ | | Luke | $$\frac{15}{5} = 3$$ | $$5$$ | $$15$$ | | Tom alone | $$5$$ | $$1$$ | $$\color{red}{5 \times 1 = 5}$$ | At the end of the first phase, the work left is $$15-5 = 10$$ acres of field.

7:15

8:15

8:25

8:53

9:53

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