I was asked how I did it by someone privately so I'm posting it here. Attached you'll see how originally I tried to use my trusty box method, but for this type of rate problem I couldn't figure out how to take the box method to the next step. So, I took the information I used by using the box method to draw it out. See the second picture for that.
The way we calculate the rate of the 6 beavers is, since they take 10 hours to do it, the rate is always the inverse of the time AS LONG AS the job they are completing it "one job". Meaning, 1/10 x 10/1 = 1. So when any problem says "to complete a job" or something of that nature, the job being (don't pick any answer choices in the verbal section that have the word "being") done means it's one job. So as long as you know any two parts of the equation, you can always get the third.
The way we calculate that the rate of one beaver is 1/60 is because we took the 1/10 (which is the rate of all six of them) and divided it by six. Had it been seven beavers we would have divided it by 7. etc...
So - since we know the rate of each beaver and we want to know how long it will take each beaver to complete one job, the time has to be the inverse of the rate, which comes to 60/1 (otherwise known as 60).
So now what... - I had been hoping that someone would be able to explain how to continue the box method here since it's awesome... but.... in the absence of that I did it the long way.
Ok - so we know that 6 beavers started the job at noon. By 1pm, 1/10 of the job was done, because they are working at a rate of 1/10 per hour, since over 10 hours they would complete the one full job. By 2pm they had completed 2/10 of the job (1/10+1/10=2/10), and so on until 6pm. Now at 6pm they added one additional beaver, so there were 7 beavers working from 6pm to 7pm. Now, you'll notice that I switched 6/10 to 42/60. Where did I get that from - so since we're adding one beaver using the format of 1/60, we need to have the 6 beavers represented with a common denominator. Easiest way to do that is to multiply both parts of the fraction by 1 in the form of 6/6, so instead of 7/10 we get 42/60. Means the same thing. Ah, but now we can add 1/60 to it - remember that the 1/60 represents the work of the additional beaver that started working at 6pm and it's 1/60 of the job was completed one hour later, at 7pm. So we take 42/60 + 1/60 and we get 43/60.
And now by 8 pm, at the "top" we have to account for the original 6 beavers continuing their 1/10 of work (now represented as 6/10), but we're adding that to 43/60 to get 49/60. And since we were told that there is an additional beaver being (again, don't use "being" in the verbal section) added per hour, there are two additional beaver's work that is added represented by 2/60. So we add 49/60 to 2/60 to get 51/60. Now the same thing as we did before - we add the work done by the same top 6 beavers 6/60 to 51/60 to get 57/60, and now since there were three beavers working, cuz remember yet another additional beaver started working at 8pm, we add 3/60 to the 57/60 to get 60/60, which is the same thing as "1", which means they completed the job @ 9pm.
Hope this long megillah helped!
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