Tom can harvest the wheat in 20 minutes. Roza can harvest the wheat in 42 minutes. If they work together, how long does it take to harvest the wheat?
tom's rate of job performance ... (1 job)/(20 min)
roza's rate of job performance ... (1 job)/(42 min)
rate together in jobs/min ... (1/20) + (1/42)
now use the relationship ...
(rate together)(time in minutes together) = 1 job completed
solve for (time in minutes together)
C'mon Josh...by yourself, it takes you 20 minutes to do the job; Ok?
So if you get ANY help whatsoever, including from slowest person around,
it will then take less than 20 minutes to do the job...see that?
Say it takes you 20 minutes to paint 20 doors.
If you get help from a slow painter who can paint only 1 door in 19 minutes,
then the job took 19 minutes, right? You only needed to paint 19...
You're absolutely right, Wilmer. This means that there are some situations in which I will not get help at all. Imagine that the slower painter can paint 1 door in 20 minutes! Then what will happen? No help, right? But, what will happen to the last door?
After you painted 19 doors, slowpoke will have painted 19/20 of the last door:
so both of you will work together to finish the 1/20 of door left.
In other words, if slowpoke is so slow he can only paint 1 square inch of a door
after 19 minutes, then quits, you still save a little time because you don't have
to paint that 1 square inch!
Let's change your problem slightly:
Art can do a job in 21 minutes. Ben can do the same job in 28 minutes.
If they work together, how long does it take to do the job?
Art, in 1 minute : 1/21 of job
Ben, in 1 minute: 1/28 of job
Art + Ben, in 1 minute : 1/21 + 1/28 = (28 + 21) / (21*28) = 49/588 = 1/12 of job
So they take 12 minutes if working together.
For a general case equation, let one worker's time = a and the other = b.
So in 1 minute, 1/a + 1/b of job gets done.
So working together = a*b / (a + b) minutes.
Let's see: 21 * 28 / (21 + 28) = 588 / 49 = 12
Similarly with your original problem:
20 * 42 / (20 + 42) = 840 / 62 = ~13.548 minutes ; or ~13 minutes 33 seconds.
Now go away and start painting doors !