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# A renowned medical store must purchase a set of n metal weights, each

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Manager
Joined: 01 Sep 2016
Posts: 180
GMAT 1: 690 Q49 V35
A renowned medical store must purchase a set of n metal weights, each  [#permalink]

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Updated on: 20 Sep 2017, 12:48
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Question Stats:

38% (02:19) correct 62% (02:32) wrong based on 87 sessions

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A renowned medical store must purchase a set of n metal weights, each weighing an integer number of grams, such that all integer weights from 1 to 300 grams (inclusive) can be made with a combination of one or more of the weights. What is the minimum number of metal weights that the medical store must purchase?

(A) 6
(B) 8
(C) 9
(D) 10
(E) 12

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we shall fight on the landing grounds,
we shall fight in the fields and in the streets,
we shall fight in the hills;
we shall never surrender!

Originally posted by bkpolymers1617 on 20 Sep 2017, 12:42.
Last edited by bkpolymers1617 on 20 Sep 2017, 12:48, edited 1 time in total.
Manager
Joined: 01 Sep 2016
Posts: 180
GMAT 1: 690 Q49 V35
Re: A renowned medical store must purchase a set of n metal weights, each  [#permalink]

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20 Sep 2017, 12:47
2
Build up the set of weights one at a time.

The pharmacy will have to have a 1-gram weight, or else it will be impossible to weigh out exactly 1 gram.

If a 2-gram weight is added, then it becomes possible to weigh out either 2 or 3 grams, so any weight from 1 to 3 grams can be weighed out.

If a 4-gram weight is then added, then weights of 4, 5, 6, and 7 grams can also be measured out; the set now covers any weight from 1 to 7 grams.

Notice the pattern: add up all existing weights to find the maximum possible measured weight. Then, the next weight to be added is the first measure that can’t yet be covered. If you havea 1-gram, 2-gram, and 4-gram weights, then the maximum measured weight is 7 grams, so the next weight to add is 8 grams.

Adding that new weight will then allow you to measure every single weight up to the new total of all four weights. Once you add an 8-gram weight, you can measure up to 15 grams total.

Continue the pattern:
-      A 16-gram weight, for weights up to 1 + 2 + 4 + 8 + 16 = 31 grams
-      A 32-gram weight, for weights up to 31 + 32 = 63 grams
-      A 64-gram weight, for weights up to 63 + 64 = 127 gram
-      A 128-gram weight, for weights up to 127 + 128 = 255 grams
-      Finally, another weight, weighing anywhere from 45 to 256 grams*

* The last one doesn’t have to follow the pattern because, this time, you only need to measure up to 300 grams total, not 255 + 256 = 511 grams. A weight of 256 grams will work, but so will any weight all the way down to the minimum needed to reach 300 (a weight of 45, because 255 + 45 = 300).

Therefore, the pharmacy needs a minimum of nine weights.

To prove that this is indeed the minimum, note that the total number of different gram weights that can theoretically be measured with a set of eight metal weights is 28 – 1 = 255. The 28 comes from the fact that each weight can be either present or absent, so there are eight separate decisions with 2 options each: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 28. Finally, subtract 1 for the case in which all eight weights are absent, in which case you aren’t measuring anything (you’re measuring a weight of 0).

The first eight weights—chosen to measure any integer weight from 1 to 255 grams—are actually optimal; no better set can be found. Therefore, a ninth weight is necessary to reach 300 grams. One weight of 45 grams, for example, will allow you to measure 300 as well as any weight between 255 and 300. (The pattern above proves this—each new weight allows you to measure any integer weights up to the combined total for all weights—but, if you’re not sure, add up the weights needed to create 256, 257, and so on until you’ve convinced yourself.)

The correct answer is (C).
_________________
we shall fight on the beaches,
we shall fight on the landing grounds,
we shall fight in the fields and in the streets,
we shall fight in the hills;
we shall never surrender!
Director
Joined: 13 Mar 2017
Posts: 713
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Energy and Utilities)
Re: A renowned medical store must purchase a set of n metal weights, each  [#permalink]

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12 Dec 2018, 21:17
bkpolymers1617 wrote:
A renowned medical store must purchase a set of n metal weights, each weighing an integer number of grams, such that all integer weights from 1 to 300 grams (inclusive) can be made with a combination of one or more of the weights. What is the minimum number of metal weights that the medical store must purchase?

(A) 6
(B) 8
(C) 9
(D) 10
(E) 12

Can we expect such question in GMAT 700 level.. It seems pretty tough question..
Any GMAT Expert please.
Senior Manager
Joined: 12 Sep 2017
Posts: 306
Re: A renowned medical store must purchase a set of n metal weights, each  [#permalink]

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13 Dec 2018, 19:09
1
Is it possible to solve it by the number of factors?

LCM: 300 = 2*2 - 3*1 - 5*2

Hence...

2+1= 3
1+1 = 2
2 + 1=3

3 + 2 + 3 + 1 (1 gr inclusive) = 9

C
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Re: A renowned medical store must purchase a set of n metal weights, each  [#permalink]

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23 Jun 2020, 04:35
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Re: A renowned medical store must purchase a set of n metal weights, each   [#permalink] 23 Jun 2020, 04:35

# A renowned medical store must purchase a set of n metal weights, each

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