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Math Help - How many men in total were under Genghis Khan in his organization?

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    How many men in total were under Genghis Khan in his organization?

    *I figured this was the best place to ask this question because it didn't seem to fit in any of the other categories*

    Genghis Khan organized his men into groups of 10 soldiers under a “leader of 10.” Ten “leaders of 10” were under a “leader of 100.” Ten “leaders of 100” were under a “leader of 1,000.” (a) If Khan had an army of 10,000 soldiers at the lowest level, how many men in total were under him in his organization? (b) If Khan had an army of 5,763 soldiers at the lowest level, how many men in total were under him in his organization?
    Assume that the groups of 10 should contain 10 if possible, but that one group at each level may need to contain fewer.

    How do I go about solving this problem? For example (a) says that Khan had an army of 10,000 soldiers. So does that mean he had 10,000 men under him? Or do I add the leaders of each group so: 10 leaders of 10,000 + 10 leaders of 1,000 + 10 leaders of 100 + 10 leaders of 10 = 40 leaders (men) under him and then the same concept for (b)?
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    Re: How many men in total were under Genghis Khan in his organization?

    Hey lamenofking.

    You should look at the branches individually and then add them all up. A branch consists of the authority tree with Ghenghis at the root and the lowest level soldier at the bottom.

    If each set of branches is the same as every other branch at that level in the tree then you can use the multiplication rule and the total number of branches will be found by multiplying the numbers of nodes at each branch. If this is not the case, then you will have to add up each branch individually and simplify where-ever possible if you can.
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    Re: How many men in total were under Genghis Khan in his organization?

    Hi,
    Maybe different phraseology would help. The army has privates, sergeants, captains, colonels, generals, etc. Each rank, after privates, in the previous list has 10 officers below him, if possible. Example: The army has 22 privates; then there are 2 sergeants each commanding 10 privates and 1 sergeant over 2 privates. That is, the sergeant count is ceiling(22/10) = 3. Here ceiling is the least integer greater than or equal to its argument.

    So the trick is to start at the bottom, count sergeants, then captains, colonels, etc.

    5,763 privates
    ceiling(5763/10)=577 sergeants
    ceiling(577/10)=58 captains
    ceiling(58/10)=6 colonels
    ceiling(6/10)=1 general

    Total in the army = 5763+577+58+6+1=6405 (assuming the highest rank general is not the khan).

    Second example:
    7,999 privates
    ceiling(7999/10)=800 sergeants
    ceiling(800/10)=80 captains
    ceiling(80/10)=8 colonels
    ceiling(8/10)=1 general
    Total 7999+800+80+8+1=8888

    Suppose there are x privates. Let o1, o2, etc be the numbers of sergeants, captains, etc. Then o1=ceiling(x/10) and for i>1, oi=ceiling(oi-1/10). The total in the army is then T(x)=x+o1+o2+...
    Clearly there are only finitely many terms in this sum. I leave it to you to find a formula for the number of terms in this sum. In this regard, the formula ceiling(ceiling(z/10)/10)=ceiling(z/102) for any z, is helpful.
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