Here I have two different people who are painting a room. When we look at the problem it says, Jessica can paint the room in 14 hours. Lily could paint the same room in 10 hours. If they work together, how long would it take them to complete the job? This one’s easy, I just add 10 and 14 right? 24 done. No, no that’s not right.
What I need to do is figure out a time that’s faster that each one of their individual times. It doesn’t make sense that it would take them 24 hours together, because it should be faster. Well let’s figure out how to do this correctly. I want to think about how much of the job each person can do working by herself.
For example Jessica in one hour does 1/14 of the job. Lily in one hour does 1/10 of the job, so whatever the sum of this fraction is represents together how much they do in one hour. Together in 1 hour. To add fractions together, we need to find the common denominator. The common denominator for 14 and 10 is 70. Multiply that guy by 7, the first guy by 5, so what I’m really doing is 5/70 plus 7/70 which is 12/70, reduce that by dividing top and bottom by 2 and I’ll get 6/35. This isn’t my answer, this just tells me that in one hour working together they can do this much of the room, 6/35 of the room. It’s a nasty fraction but a lot of times in the real world, you don’t get pretty numbers, pretty meaning integers of halves or quarters or whatever.
Okay, so I’m going to write this in another equation now where I know how much they do together in one hour times however many hours they’re working, it’s what I’m solving for is going to be equal to 1 total job. To solve for x multiple by the reciprocal of this fraction and I get x equals 35/6 of an hour. If you do that on your calculator I believe you get 5.8, so I’m going to rewrite that up here. X equals 5.83 hours. That’s how long it takes them together and that makes a lot more sense than the silly answer I put in the beginning of 24 hours.
So guys when you have these work problems, just be aware you’re going to have to deal with fractions, common denominators and possibly something that’s kind of unattractive. When it comes to numbers we have attractive numbers and unattractive numbers. That one I mean unattractive because it’s a non-terminating decimal and that’s kind of tricky.
So when you get a problem like this, just try to figure out how much each could do individually by his or herself, add those together to see how much they do together in one hour and then set that into this equation. Amount done in one hour times sum number of hours is equal to one total job.
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