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# Halfway through a 100-shot archery tournament, Chelsea leads by 50 poi

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Halfway through a 100-shot archery tournament, Chelsea leads by 50 poi  [#permalink]

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21 Mar 2019, 06:10
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Difficulty:

95% (hard)

Question Stats:

37% (02:51) correct 63% (02:15) wrong based on 30 sessions

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Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea’s next n shots are bulleyes she will be guaranteed victory. What is the minimum value for n?

(A) 38

(B) 40

(C) 42

(D) 44

(E) 46
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Re: Halfway through a 100-shot archery tournament, Chelsea leads by 50 poi  [#permalink]

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21 Mar 2019, 08:30
1
She has a 50 point lead, and on the final fifty shots, the person in second place can score at most 500 points (10 points per shot). So as long as she scores more than 450 points on her final fifty shots, she will be certain to win.

If she gets n consecutive bullseyes, scoring 10 points each, she will get 10n points in total. Then on her final 50-n shots, we know she also scores at least 4 points, so she will get at least another 4(50 - n) = 200 - 4n points. So she will score at least 10n + 200 - 4n = 6n + 200 points. This needs to be greater than 450, so

6n + 200 > 450
6n > 250
n > 41 + 2/3

and the smallest possible integer value of n is 42.
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Re: Halfway through a 100-shot archery tournament, Chelsea leads by 50 poi  [#permalink]

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23 Mar 2019, 01:31
She has a 50-point lead, so count the max number of times she can miss the bullseye while her opponent does hit the bullseye.
With every bullseye miss she loses a 6 point lead, because she always gets at least 4 (10 - 4 = 6).
Miss 1: 50 - 6 = 44
2: 38
3: 32
4: 26
5: 20
6: 14
7: 8
8: 2
If she misses the next bullseye she will be behind so this is the max number of misses.
50 shots left - 8 = 42
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Re: Halfway through a 100-shot archery tournament, Chelsea leads by 50 poi  [#permalink]

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30 Mar 2019, 12:02
My solution
Max points that can be lost on each shot is 6 points. Lead is 50. So for 8 shots they can afford to not hit bulls eye while opponents hit bullseye. So 42 guarantees victory

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Re: Halfway through a 100-shot archery tournament, Chelsea leads by 50 poi  [#permalink]

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30 Mar 2019, 23:46
IanStewart wrote:
She has a 50 point lead, and on the final fifty shots, the person in second place can score at most 500 points (10 points per shot). So as long as she scores more than 450 points on her final fifty shots, she will be certain to win.

If she gets n consecutive bullseyes, scoring 10 points each, she will get 10n points in total. Then on her final 50-n shots, we know she also scores at least 4 points, so she will get at least another 4(50 - n) = 200 - 4n points. So she will score at least 10n + 200 - 4n = 6n + 200 points. This needs to be greater than 450, so

6n + 200 > 450
6n > 250
n > 41 + 2/3

and the smallest possible integer value of n is 42.

Why did you assume that she scored only 4 points on 50-n shots? The question says atleast 4. it could be 4 or 8. Any specific reason to select 4?
Re: Halfway through a 100-shot archery tournament, Chelsea leads by 50 poi   [#permalink] 30 Mar 2019, 23:46
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