In a sports competition, there were M medals awarded on N successive days, where N is greater than 1.
- On the first day: 1 Medal and 1/9 of the remaining (M-1) medals were awarded.
- On the second day: 2 Medals and 1/9 of the now remaining medals were awarded, ...and, so on.
- On the Nth and the last day: the remaining N medals were awarded.
How many days did the contest last and how many medals were awarded altogether?
Provide valid reasoning for your answers.
Consider the penultimate day, and let the number of medals remaining on that day be R.
Then on the last day, N=8R/9
Since 8 and 9 have no common factors, R=9 seems like an obvious substitution. Note that if R>9, say 18, then the result on the last day is greater than N, a contradiction, while if R<9, N has a fractional part, so this solution, if valid, is unique.
If therefore a solution exists, then N=8 and R=9.
To check this, assume that 8 medals were awarded on the Nth (8th) day, with 0 medals remaining.
On the previous (7th) day, (N-1)+1/9(R) medals were awarded, for a total of 8, with 8 medals remaining.
Note that for the 6th day, we can gross up the remaining medals by 9/8 to achieve the number before distribution.
On the 6th day, (N-2)+1/9(9/8*(8+8)) medals were awarded, for a total of 8, with 16 medals remaining.
On the 5th day, (N-3)+1/9(9/8*(8+8+8)) medals were awarded, for a total of 8, with 24 medals remaining.
and so on.
So the same number of medals (8) were distributed each day for 8 days, giving a total of 64.
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Posted by broll
on 2022-04-30 23:58:17 |
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