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Question 788871: Sue is packing boxes ,she packed 55 boxes in 5 hours. Since she is getting better at it, each hour she packs 2 more than the previous hour. How many boxes did she pack in the fourth hour?
Regards
One stumped nana trying to help grandson
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! THE STORY:
I read the problem as meaning that Sue packed a total of 55 boxes during hours 1 through 5, and the question is how many boxes did Sue pack in the fourth of those 5 hours.
In those 5 hours Sue has been packing boxes at an average speed of  boxes per hour.
Since her packing speed increases at a constant rate, her average rate for the 5 hours is the same as her rate for the third hour (the one in the middle of the 5).
So, during the third hour she packed 11 boxes.
The 2 extra boxes packed in the fourth hour (for a total of  boxes) compensate for the fact that during the second hour Sue she had packed 2 boxes less than 11.
Also, compared to the third hour, the 4 extra boxes packed during the fifth hour compensate for the 4 fewer boxes packed during the first hour.
So, Sue packed 7 boxes, the first hour, 9 the second hour, 11 the third hour,  the fourth hour, and 15 the fifth hour.
It all adds up.
HOW TO SHOW THE WORK:
If only the final result is required, we are done.
If this is an elementary, or middle school problem, probably they expect the student to use "guess and check", taking a guess on how many boxes were packed the first hour, and working from there to find a 5-hour total. If the total is not right, you have to make a new guess and calculate a new total. If the total is too low, the first hour guess has to be increased; if the total is too high, the first hour guess has to be decreased. A table can be made showing the guesses and the results. When you guess that 7 boxes were packed the first hour , you get a 5-hour total of 55 boxes, and a fourth hour count of 13 boxes (the answer to the problem).
If this is an advanced high school math problem, and the student is studying arithmetic sequences (also called arithmetic progressions in some places), the teacher probably expects to see a bunch of formulas.
The arithmetic sequence terms would be given by
 = first term = numbers of boxes packed thr first hour
 = common difference = difference in the numbers of boxes packed from one hour to the next.
 = boxes backed the nth hour
 = sum of the first n terms = total number of boxes packed in n hours.
In this case,  ,  ,  and we have to solve for the equation
Then, we plug  , and into to find the number of boxes packed the fourth hour.
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