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# Bill owns a large collection of fishing lures consisting of

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Joined: 11 May 2013
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Bill owns a large collection of fishing lures consisting of  [#permalink]

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Updated on: 08 Oct 2013, 00:48
5
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Difficulty:

55% (hard)

Question Stats:

61% (01:24) correct 39% (01:47) wrong based on 247 sessions

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Bill owns a large collection of fishing lures consisting of small, medium, and large lures that weigh 3, 4, and 5 grams each, respectively. If the product of the lure weights that Bill sold to his friend is 216,000 grams, how many medium lures did he sell?

A. 6
B. 5
C. 4
D. 3
E. 2

Analyze the Question:

Bill sold some small, medium, and large lures to his friend. The product of the weights of the lures is 216,000 grams. We know the weights of the individual lures, but we do not know how many of each type he sold. We can factor the product to find the number of 4-gram weights sold. The answer choices are very close together, and these values can help guide the solution process.

We will determine the number of 4-gram lures Bill sold to his friend by determining the number of factors of 4 there are in 216,000.

Approach Strategically:

Number properties are the key to solving this problem quickly. Since 3, 4, and 5 do not have any common factors with each other, the number of 4-gram lures Bill sold must be the number of factors of 4 in 216,000. Dividing 216,000 by 4 yields 54,000, and 54,000 is still a multiple of 4 due to its last two digits of 00. Dividing 54,000 by 4 yields 13,500, and 13,500 is still a multiple of 4 due to its last two digits of 00. Dividing 13,500 by 4 yields 3,375, and 3,375 is not a multiple of 4 as its last two digits, 75, are not divisible by 4. That’s three factors of 4 for 216,000, so Bill sold three 4-gram lures to his friend.

The correct answer is Choice (D).

We can factor 216,000 completely using the given weights and see that 216,000 = 3^3× 4^3× 5^3. This is the only possible way to get 216,000 from 3, 4, and 5, so we know that three of each of these lures were sold.

Originally posted by jabronyo on 07 Oct 2013, 16:41.
Last edited by Bunuel on 08 Oct 2013, 00:48, edited 1 time in total.
RENAMED THE TOPIC.
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Re: How many medium lures did Bill sell?  [#permalink]

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07 Oct 2013, 18:21

You will get (2)with exponent 6.

Which is nothing but 4 with exponent 3.

Thanks
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Re: How many medium lures did Bill sell?  [#permalink]

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07 Oct 2013, 19:13
total weight = 216000

weight of medium lure - 4 gm
by factoring we get 5*5*5*3*3*3*4*4*4
THUS MEDIUM LURE = 3
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Re: How many medium lures did Bill sell?  [#permalink]

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07 Oct 2013, 20:51
2
1. Let the number of small, medium and large lures sold be x,y and z resp.
2. The product of the weights of the lures is 3^x * 4^y * 5^z
3. But it is also given the product is 216,000 = 3^3 * 4^3 * 5^3 by factorization.
4. So y=3.
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Re: Bill owns a large collection of fishing lures consisting of  [#permalink]

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24 Dec 2014, 16:10
Hi All,

Everyone who posted in this thread seems comfortable with prime factorization, but for anyone coming across this post who isn't completely comfortable with the "math", here's how prime factorization "works" - every positive integer (except the number 1) is either a prime number OR the product of a bunch of prime numbers.

For example, 2 and 3 are both prime numbers, but 4 is the product of 2 and 2.

Here, we're given a BIG number and asked to figure out the numbers that make up its PRODUCT. Since the numbers 3 and 5 are prime and 4 = 2^2, this is a big "clue" that we can use prime factorization to get to the correct answer.

So, let's prime factor 216,000

You can break this number up any way you choose, but I usually look for a logical "split"...

(216)(1,000)

(216) =
(4)(54)
(4)(6)(9)
(4)(2)(3)(3)(3)

**NOTICE that we have one 4 and one 2**

(1,000)
(10)(10)(10)
(2)(5)(2)(5)(2)(5)

**NOTICE that we have three 2s***

We now have enough information to determine how many 3s, 4s and 5s make up 216,000:

Three 3s
Three 5s
We have one 4 and four 2s. The 2s can be "paired up" to create a 4, so we have
Three 4s

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Re: Bill owns a large collection of fishing lures consisting of  [#permalink]

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31 Dec 2014, 00:33
1
216000 = 216 * 1000

216 is a perfect cube of 6, 1000 is a perfect cube of 10

$$216000 = 6^3 * 10^3$$

$$= 2^3 * 3^3 * 2^3 * 5^3$$

$$= (2*2)^3 * 3^3 * 5^3$$

$$= 4^3 * 3^3 * 5^3$$

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Re: Bill owns a large collection of fishing lures consisting of  [#permalink]

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10 Jan 2015, 13:13
Just as an equation, using sort of the unknown mupltiplier approach:

(3^x)*(4^x)*(5^x)= 216.000. Here you need the prime factorization of 216.000, which I will not show as it has been shown above.

Then you get this:
(3^x)*(4^x)*(5^x) = (3^3)*(4*3)*(5^3), so 4^x = 4*3 and x=3.

Does this make any sense?
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Re: Bill owns a large collection of fishing lures consisting of  [#permalink]

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09 Oct 2016, 11:11
jabronyo wrote:
Bill owns a large collection of fishing lures consisting of small, medium, and large lures that weigh 3, 4, and 5 grams each, respectively. If the product of the lure weights that Bill sold to his friend is 216,000 grams, how many medium lures did he sell?

A. 6
B. 5
C. 4
D. 3
E. 2

pretty straight forward question...find the prime factorization of 216,000 and tell the exponent number for 4 or 2^2.

first step:
216,000 = 216 * 1000
1000 = 10^3 or 5^3 * 2^3.
216 = 2 * 108 = 2*2* 54
54 = 2*27 = 2*3^3.
total, we have:
2^3(from 1000) * 5^3(from 1000) * 2^3(from 216) * 3^3(from 2016)
combine:
2^6 * 5^3 * 3^3
we can rewrite 2^6 as 4^3.
since we are asked for the exponent for 4, the answer should be 3 - D
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Bill owns a large collection of fishing lures consisting of  [#permalink]

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17 Oct 2017, 20:50
Bill owns a large collection of fishing lures consisting of small, medium, and large lures that weigh 3, 4, and 5 grams each, respectively. If the product of the lure weights that Bill sold to his friend is 216,000 grams, how many medium lures did he sell?

A. 6
B. 5
C. 4
D. 3
E. 2

the product of one set of three lures--s,m,l--will=3*4*5=60 grams
the product of three sets of three lures will=60^3=216,000 grams
in three sets there will be 3 medium lures
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Re: Bill owns a large collection of fishing lures consisting of  [#permalink]

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19 Oct 2017, 10:20
jabronyo wrote:
Bill owns a large collection of fishing lures consisting of small, medium, and large lures that weigh 3, 4, and 5 grams each, respectively. If the product of the lure weights that Bill sold to his friend is 216,000 grams, how many medium lures did he sell?

A. 6
B. 5
C. 4
D. 3
E. 2

We can break down 216,000 into primes:

216,000 = 216 x 1000 = 6^3 x 10^3 = 2^3 x 3^3 x 2^3 x 5^3 = 2^6 x 3^3 x 5^3

However, we really want to break the number 216,000 into factors of 3, 4, and 5 to match the weights of the lures that he sold. Thus, we should rewrite the product as:

216,000 = 2^6 x 3^3 x 5^3 = 4^3 x 3^3 x 5^3

Since each medium lure weighed 4 grams, he sold 3 medium lures.

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Re: Bill owns a large collection of fishing lures consisting of &nbs [#permalink] 19 Oct 2017, 10:20
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