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On this year's Westchester basketball team, the players are all either

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On this year's Westchester basketball team, the players are all either  [#permalink]

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New post 05 Sep 2018, 09:43
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On this year's Westchester basketball team, the players are all either 5,7,or 11 years of age. If the product of ages of the players on the team is 18,865 , then what is the probability that a randomly selected team member will not be 7


A \(\frac{3}{7}\)
B \(\frac{2}{5}\)
C \(\frac{16}{37}\)
D \(\frac{3}{5}\)
E \(\frac{49}{55}\)
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On this year's Westchester basketball team, the players are all either  [#permalink]

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New post Updated on: 06 Sep 2018, 08:42
2
This is a time consuming but not impossible question. Here is how I would solve this question.

Understand the Question - Behind this word problem lurks a pretty standard Number Properties / Primes and divisibility question. The core skill being tested is the ability to prime factor a number, in this case 18,865. The good news is that they tell you the three factors - 5, 7 and 11. One of these numbers, 5, is easy to factor out. One of these numbers, 11, has a little remembered rule. Lastly, one of these numbers, 7, has a super complicated algorithm that is better off ignored. So, a simple restatement of this question is "How many factors of 5, 7, and 11 are in 18,865?"

Understand the Answer Choices - The denominator of the whichever answer choice is correct will at maximum be equal to the total number of 5s, 7s, and 11s in 18,865 - the numerator will be the number of 5s and 11s. This is from the probability equation = (total number of desired outcomes)/(total number of outcomes). This makes answer choices C and E pretty unlikely as the denominators are way too high. Eliminate.

Plan - what's the fastest way to attack this question? Well, you know by the question that there is at least one 5, 7, and 11. So, dividing by these numbers is a reasonable way to start. I would divide by 5 or 11 first as these are faster. If you happen to be slow at division, guessing among A, B, and D is also reasonable.

Solve - 18865 / 5 = 3,773. No more factors of 5 (Do you know why?)(Does not end in 0 or 5)
3,773 / 11 = 343 No more factors of 11 (Do you know why?) (the outside numbers do not add to the inside number)
343 / 7 = 49
49 = 7 * 7

So the prime factorization of 18,865 is 5*7*7*7*11
Eliminate A as 7 will not be the denominator
Desired outcomes = 2 (one 5 and one 11)
Total Outcomes = 5
B is correct

Jayson Beatty
Indigo Prep
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Jayson Beatty
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http://www.indigoprep.com


Originally posted by jaysonbeatty12 on 05 Sep 2018, 10:18.
Last edited by jaysonbeatty12 on 06 Sep 2018, 08:42, edited 1 time in total.
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Re: On this year's Westchester basketball team, the players are all either  [#permalink]

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New post 06 Sep 2018, 00:09
1
jaysonbeatty12 wrote:
This is a time consuming but not impossible question. Here is how I would solve this question.

Understand the Question - Behind this word problem lurks a pretty standard Number Properties / Primes and divisibility question. The core skill being tested is the ability to prime factor a number, in this case 18,865. The good news is that they tell you the three factors - 5, 7 and 11. One of these numbers, 5, is easy to factor out. One of these numbers, 11, has a little remembered rule. Lastly, one of these numbers, 7, has a super complicated algorithm that is better off ignored. So, a simple restatement of this question is "How many factors of 5, 7, and 11 are in 18,865?"

Understand the Answer Choices - The denominator of the whichever answer choice is correct will at maximum be equal to the total number of 5s, 7s, and 11s in 18,865 - the numerator will be the number of 5s and 11s. This is from the probability equation = (total number of desired outcomes)/(total number of outcomes). This makes answer choices C and E pretty unlikely as the denominators are way too high. Eliminate.

Plan - what's the fastest way to attack this question? Well, you know by the question that there is at least one 5, 7, and 11. So, dividing by these numbers is a reasonable way to start. I would divide by 5 or 7 first as these are faster. If you happen to be slow at division, guessing among A, B, and D is also reasonable.

Solve - 18865 / 5 = 3,773. No more factors of 5 (Do you know why?)(Does not end in 0 or 5)
3,773 / 11 = 343 No more factors of 11 (Do you know why?) (the outside numbers do not add to the inside number)
343 / 7 = 49
49 = 7 * 7

So the prime factorization of 18,865 is 5*7*7*7*11
Eliminate A as 7 will not be the denominator
Desired outcomes = 2 (one 5 and one 7)
Total Outcomes = 5
B is correct

Jayson Beatty
Indigo Prep



The solution is correct , there is however a small typo , it should be 11 instead of 7
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Re: On this year's Westchester basketball team, the players are all either  [#permalink]

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New post 06 Sep 2018, 09:57
2
CounterSniper wrote:
On this year's Westchester basketball team, the players are all either 5,7,or 11 years of age. If the product of ages of the players on the team is 18,865 , then what is the probability that a randomly selected team member will not be 7?

A \(\frac{3}{7}\)
B \(\frac{2}{5}\)
C \(\frac{16}{37}\)
D \(\frac{3}{5}\)
E \(\frac{49}{55}\)

\(? = P\left( {{\text{choose}}\,{\text{at}}\,{\text{random}}\,\,{\text{not}}\,{\text{a}}\,\,7{\text{y}}\,{\text{player}}} \right)\)

There are only 2 real "problems" here:

1. To factorize 18,865 quickly to get:
\(18865 = 5 \cdot {7^3} \cdot 11\)

2. To know how many players are in a basketball team: 5 (*) so that we may conclude that:
(*) https://www.google.com.br/search?q=numb ... e&ie=UTF-8

There are one 5y , three 7y and one 11y players.

Answer: 2/5.

To factorize 18,865 I suggest:

(a) Obviously divisible by 5, to get (in less than a minute):

\(\frac{{18865}}{5} = \frac{{18 \cdot 1000 + 500 + 300 + 50 + 15}}{5} = 18 \cdot 200 + 100 + 60 + 10 + 3\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,18865 = 5 \cdot 3773\)

(b) 3773 is OBVIOUSLY divisible by 11 (because 3-7+7-3 = 0) :: check this: https://www.math.hmc.edu/funfacts/ffiles/10013.5.shtml

Hence:

\(\frac{{3773}}{{11}} = \frac{{3300 + 33 + 44 \cdot 10}}{{11}} = 300 + 3 + 40 = 343\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,3773 = 11 \cdot 343\,\,\,\)

(c) Now the hard part (how would we imagine 343 is divisible by 7?)... but the examiner gave us the hint(only 5y, 11y... and 7y players!):

Hence:

\(\frac{{343}}{7} = \frac{{350 - 7}}{7} = 49\,\,\,\,\,\, \Rightarrow \,\,\,\,\,3773 = 7 \cdot 49 = {7^3}\,\,\)

Finally, let´s group all together:

\(18865 = 5 \cdot 11 \cdot {7^3}\)


This powerful breaking numbers technique is part of our method, by the way!


This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.
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Re: On this year's Westchester basketball team, the players are all either  [#permalink]

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New post 11 Sep 2018, 07:56
This is a time consuming but not impossible question. Here is how I would solve this question.

Understand the Question - Behind this word problem lurks a pretty standard Number Properties / Primes and divisibility question. The core skill being tested is the ability to prime factor a number, in this case 18,865. The good news is that they tell you the three factors - 5, 7 and 11. One of these numbers, 5, is easy to factor out. One of these numbers, 11, has a little remembered rule. Lastly, one of these numbers, 7, has a super complicated algorithm that is better off ignored. So, a simple restatement of this question is "How many factors of 5, 7, and 11 are in 18,865?"

Understand the Answer Choices - The denominator of the whichever answer choice is correct will at maximum be equal to the total number of 5s, 7s, and 11s in 18,865 - the numerator will be the number of 5s and 11s. This is from the probability equation = (total number of desired outcomes)/(total number of outcomes). This makes answer choices C and E pretty unlikely as the denominators are way too high. Eliminate.

Plan - what's the fastest way to attack this question? Well, you know by the question that there is at least one 5, 7, and 11. So, dividing by these numbers is a reasonable way to start. I would divide by 5 or 11 first as these are faster. If you happen to be slow at division, guessing among A, B, and D is also reasonable.

Solve - 18865 / 5 = 3,773. No more factors of 5 (Do you know why?)(Does not end in 0 or 5)
3,773 / 11 = 343 No more factors of 11 (Do you know why?) (the outside numbers do not add to the inside number)
343 / 7 = 49
49 = 7 * 7

So the prime factorization of 18,865 is 5*7*7*7*11
Eliminate A as 7 will not be the denominator
Desired outcomes = 2 (one 5 and one 11)
Total Outcomes = 5
B is correct

Jayson Beatty
Indigo Prep

I understood the factorization part. I did not understand the part about why a certain number can/cannot be in the numerator.
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On this year's Westchester basketball team, the players are all either  [#permalink]

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New post 11 Sep 2018, 10:06
shardul171 wrote:
This is a time consuming but not impossible question. Here is how I would solve this question.

Understand the Question - Behind this word problem lurks a pretty standard Number Properties / Primes and divisibility question. The core skill being tested is the ability to prime factor a number, in this case 18,865. The good news is that they tell you the three factors - 5, 7 and 11. One of these numbers, 5, is easy to factor out. One of these numbers, 11, has a little remembered rule. Lastly, one of these numbers, 7, has a super complicated algorithm that is better off ignored. So, a simple restatement of this question is "How many factors of 5, 7, and 11 are in 18,865?"

Understand the Answer Choices - The denominator of the whichever answer choice is correct will at maximum be equal to the total number of 5s, 7s, and 11s in 18,865 - the numerator will be the number of 5s and 11s. This is from the probability equation = (total number of desired outcomes)/(total number of outcomes). This makes answer choices C and E pretty unlikely as the denominators are way too high. Eliminate.

Plan - what's the fastest way to attack this question? Well, you know by the question that there is at least one 5, 7, and 11. So, dividing by these numbers is a reasonable way to start. I would divide by 5 or 11 first as these are faster. If you happen to be slow at division, guessing among A, B, and D is also reasonable.

Solve - 18865 / 5 = 3,773. No more factors of 5 (Do you know why?)(Does not end in 0 or 5)
3,773 / 11 = 343 No more factors of 11 (Do you know why?) (the outside numbers do not add to the inside number)
343 / 7 = 49
49 = 7 * 7

So the prime factorization of 18,865 is 5*7*7*7*11
Eliminate A as 7 will not be the denominator
Desired outcomes = 2 (one 5 and one 11)
Total Outcomes = 5
B is correct

Jayson Beatty
Indigo Prep

I understood the factorization part. I did not understand the part about why a certain number can/cannot be in the numerator.



Once we are done with factorization
we are left with finding the probability

what do we need ?
probability that a randomly selected team member will not be 7

total possible outcomes = number of (prime factors of 18865=5*7*7*7*11) = 5
required outcome = total possible outcomes - number of 7's =2

p(not being 7) = 2/5
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On this year's Westchester basketball team, the players are all either &nbs [#permalink] 11 Sep 2018, 10:06
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