Mira is making telescopes, each consisting of 2 lenses, 1 tube, and 1 eyepiece. Lenses can be purchased only in packs of 50, tubes only in packs of 10, and eyepieces only in packs of 30. However, half of the lenses in each pack are not usable for telescopes. If all parts are used only for the telescopes, what is the minimum number of lenses Mira must purchase to make a set of telescopes with no leftover components other than the unusable lenses?
A. 75
B. 150
C. 300
D. 600
E. 7,500
How will i solve this kind of problem?
OA D
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Mira is making telescopes, each consisting of 2 lenses
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A pack of 50 lenses (ignoring unusable for now) can make 50/2 = 25 telescopes.lheiannie07 wrote:Mira is making telescopes, each consisting of 2 lenses, 1 tube, and 1 eyepiece. Lenses can be purchased only in packs of 50, tubes only in packs of 10, and eyepieces only in packs of 30. However, half of the lenses in each pack are not usable for telescopes. If all parts are used only for the telescopes, what is the minimum number of lenses Mira must purchase to make a set of telescopes with no leftover components other than the unusable lenses?
A. 75
B. 150
C. 300
D. 600
E. 7,500
How will i solve this kind of problem?
OA D
A pack of 10 tubes can make 10/1 = 10 telescopes.
A pack of 30 eyepieces can make 30/1 = 30 telescopes.
To find a situation such that all the lenses (ignoring unusable for now), all the tubes, and all the eyepieces are uses, we must take the LCM of 25, 10 and 30 to reach the minimum number of telescopes.
LCM of 25, 10 and 30 = 150 telescopes
The number of packs of lenses needed to make 150 telescopes = 150/25 = 6
Since half the lenses are unusable, we Mira must by 12 packs; thus, the number of lenses needed = 12*50 = 600.
The correct answer: D
Needless to state that to make 150 telescopes, 150/10 = 15 packs of tubes and 150/30 = 5 packs of eyepieces will be consumed leaving nothing left.
Hope this helps!
-Jay
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Ratio of lenses:tubes:eyepieces
2L:1T:1E
Quantities in half-packs
25:5:15
However, 25/2 leaves a remainder, so the number of packs of Lenses needs to EVEN.
Therefore we can omit answer A (75).
If 150 lenses are bought, that will provide 75 lenses which is not even either, so we can omit answer B.
If 300 lenses are bought, that will provide 150 useable lens to make 75 telescopes.
75 is a multiple of both 5 (half of 10) and 15 (half of 30), so the answer is C.
2L:1T:1E
Quantities in half-packs
25:5:15
However, 25/2 leaves a remainder, so the number of packs of Lenses needs to EVEN.
Therefore we can omit answer A (75).
If 150 lenses are bought, that will provide 75 lenses which is not even either, so we can omit answer B.
If 300 lenses are bought, that will provide 150 useable lens to make 75 telescopes.
75 is a multiple of both 5 (half of 10) and 15 (half of 30), so the answer is C.
A good argument, except that surely only 75 telescopes are required:Jay@ManhattanReview wrote:A pack of 50 lenses (ignoring unusable for now) can make 50/2 = 25 telescopes.lheiannie07 wrote:Mira is making telescopes, each consisting of 2 lenses, 1 tube, and 1 eyepiece. Lenses can be purchased only in packs of 50, tubes only in packs of 10, and eyepieces only in packs of 30. However, half of the lenses in each pack are not usable for telescopes. If all parts are used only for the telescopes, what is the minimum number of lenses Mira must purchase to make a set of telescopes with no leftover components other than the unusable lenses?
A. 75
B. 150
C. 300
D. 600
E. 7,500
How will i solve this kind of problem?
OA D
A pack of 10 tubes can make 10/1 = 10 telescopes.
A pack of 30 eyepieces can make 30/1 = 30 telescopes.
To find a situation such that all the lenses (ignoring unusable for now), all the tubes, and all the eyepieces are uses, we must take the LCM of 25, 10 and 30 to reach the minimum number of telescopes.
LCM of 25, 10 and 30 = 150 telescopes
The number of packs of lenses needed to make 150 telescopes = 150/25 = 6
Since half the lenses are unusable, we Mira must by 12 packs; thus, the number of lenses needed = 12*50 = 600.
The correct answer: D
Needless to state that to make 150 telescopes, 150/10 = 15 packs of tubes and 150/30 = 5 packs of eyepieces will be consumed leaving nothing left.
Hope this helps!
-Jay
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75 telescopes can be made from 150 lenses (half of 300 (6 packs of 50)), 75 tubes (half 150 (15 packs of 10) and 75 eyepieces (half of 150 (5 packs of 30)).
So only 300 lenses need to be purchased (ANSWER C).
I see the error of my ways! In misreading the question, I thought that all components were only halved due to being unusable. Only the lenses are 'half useable'. Therefore the answer is 600.
DrMaths wrote:Ratio of lenses:tubes:eyepieces
2L:1T:1E
Quantities in half-packs
25:5:15
However, 25/2 leaves a remainder, so the number of packs of Lenses needs to EVEN.
Therefore we can omit answer A (75).
If 150 lenses are bought, that will provide 75 lenses which is not even either, so we can omit answer B.
If 300 lenses are bought, that will provide 150 useable lens to make 75 telescopes.
75 is a multiple of both 5 (half of 10) and 15 (half of 30), so the answer is C.
I see the error of my ways! In misreading the question, I thought that all components were only halved due to being unusable. Only the lenses are 'half useable'. Therefore the answer is 600.DrMaths wrote:A good argument, except that surely only 75 telescopes are required:Jay@ManhattanReview wrote:A pack of 50 lenses (ignoring unusable for now) can make 50/2 = 25 telescopes.lheiannie07 wrote:Mira is making telescopes, each consisting of 2 lenses, 1 tube, and 1 eyepiece. Lenses can be purchased only in packs of 50, tubes only in packs of 10, and eyepieces only in packs of 30. However, half of the lenses in each pack are not usable for telescopes. If all parts are used only for the telescopes, what is the minimum number of lenses Mira must purchase to make a set of telescopes with no leftover components other than the unusable lenses?
A. 75
B. 150
C. 300
D. 600
E. 7,500
How will i solve this kind of problem?
OA D
A pack of 10 tubes can make 10/1 = 10 telescopes.
A pack of 30 eyepieces can make 30/1 = 30 telescopes.
To find a situation such that all the lenses (ignoring unusable for now), all the tubes, and all the eyepieces are uses, we must take the LCM of 25, 10 and 30 to reach the minimum number of telescopes.
LCM of 25, 10 and 30 = 150 telescopes
The number of packs of lenses needed to make 150 telescopes = 150/25 = 6
Since half the lenses are unusable, we Mira must by 12 packs; thus, the number of lenses needed = 12*50 = 600.
The correct answer: D
Needless to state that to make 150 telescopes, 150/10 = 15 packs of tubes and 150/30 = 5 packs of eyepieces will be consumed leaving nothing left.
Hope this helps!
-Jay
_________________
Manhattan Review GMAT Prep
Locations: New York | Singapore | Doha | Lausanne | and many more...
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75 telescopes can be made from 150 lenses (half of 300 (6 packs of 50)), 75 tubes (half 150 (15 packs of 10) and 75 eyepieces (half of 150 (5 packs of 30)).
So only 300 lenses need to be purchased (ANSWER C).
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We can PLUG IN THE ANSWERS, which represent the minimum number of lenses Mira must purchase if there are to be no leftover components.lheiannie07 wrote:Mira is making telescopes, each consisting of 2 lenses, 1 tube, and 1 eyepiece. Lenses can be purchased only in packs of 50, tubes only in packs of 10, and eyepieces only in packs of 30. However, half of the lenses in each pack are not usable for telescopes. If all parts are used only for the telescopes, what is the minimum number of lenses Mira must purchase to make a set of telescopes with no leftover components other than the unusable lenses?
A. 75
B. 150
C. 300
D. 600
E. 7,500
Since the question stem asks for the MINIMUM number of lenses that must be purchased, start with the smallest answer choice.
A: 75
Since lenses come in packs of 50, the minimum number of lenses must be a MULTIPLE OF 50.
Eliminate A.
B: 150
Since only 1/2 the lenses are usable, the number of usable lenses = (1/2)(150) = 75.
Not viable:
Since each telescope requires 2 lenses, the total number of usable lenses must be EVEN.
Eliminate B.
C: 300
Since only 1/2 the lenses are usable, the number of usable lenses = (1/2)(300) = 150.
Since each telescope requires 2 lenses, 150 usable lenses are sufficient to make 75 telescopes.
Not viable.
Since tubes come in packs of 10, the number of telescopes must be a MULTIPLE OF 10.
D: 600
Since only 1/2 the lenses are usable, the number of usable lenses = (1/2)(600) = 300.
Since each telescope requires 2 lenses, 300 usable lenses are sufficient to make 150 telescopes.
This works.
To build 150 telescopes, Mira must purchase 15 packs of tubes (since each pack contains 10 tubes) and 5 packs of eyepieces (since each pack contains 30 eyepieces).
The correct answer is D.
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Let's assume that Mira is making n telescopes, thus she need 2n lenses, n tubes and n eyepieces. Let's also assume that she bought a packs of lenses, b packs of tubes and c packs of eyepieces, thus she bought 50a lenses, 10b tubes and 30c eyepieces.lheiannie07 wrote:Mira is making telescopes, each consisting of 2 lenses, 1 tube, and 1 eyepiece. Lenses can be purchased only in packs of 50, tubes only in packs of 10, and eyepieces only in packs of 30. However, half of the lenses in each pack are not usable for telescopes. If all parts are used only for the telescopes, what is the minimum number of lenses Mira must purchase to make a set of telescopes with no leftover components other than the unusable lenses?
A. 75
B. 150
C. 300
D. 600
E. 7,500
Since she can only use half of the lenses, she used 25a lenses. So we have:
2n = 25a, n = 10b and n = 30c
Since n, a, b, and c have to be integers, we see that the smallest value of a is 2. If that is the case, then n will be 25, but then b will be 2.5, which is not an integer. So a can't be 2.
The next smallest value of a is 4. If that is the case, then n will be 50 and b will be 5, but then c will be 5/3, which is not an integer. So a can't be 4.
Notice that a has to be even in order for n to be an integer and we can keep trying values for a (and make sure that b and c will be integers also):
If a = 6, then n = 75 and b = 7.5.
If a = 8, then n = 100 and c = 10/3.
If a = 10, then n = 125 and b = 12.5.
If a = 12, then n = 150 and b = 15 and c = 5.
We've founded the value of a (a = 12) so that n, b and c are all integers. Thus the minimum number of lenses she's used is 25 x 12 = 300. However, she needed to buy twice as many lenses, or 600 lenses, since half of them are unusable.
Alternate Solution:
Let's test each answer choice, starting from the smallest:
Answer Choice A: 75 lenses cannot be purchased since lenses come in packs of 50
Answer Choice B: If 150 lenses are purchased, only 75 of them can be used for making a telescope and since each telescope uses two lenses, one usable lens will be leftover.
Answer Choice C: If 300 lenses are purchased, only 150 of them can be used for making a telescope and since each telescope uses two lenses, a total of 75 telescopes can be made. But to make 75 telescopes, Mira will
have to buy at least 8 packs of tubes for a total of 8 x 10 = 80 tubes; 5 of which will be leftover.
Answer Choice D: If 600 lenses are purchased, only 300 of them can be used for making a telescope and since each telescope uses two lenses, a total of 150 telescopes can be made. To make 150 telescopes, Mira can purchase 15 packs of tubes and 5 packs of eyepieces with no leftover parts besides unusable eyepieces.
Answer: D
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Hi lheiannie07,
This question is perfect for TESTing THE ANSWERS (as Mitch has shown in his solution). It can also be solved with "math" by focusing on MULTIPLES of the numbers involved.
We're told that we can buy:
-Lenses in packs of 50
-Tubes in packs of 10
-Eyepieces in packs of 30
There are some additional details to consider:
-Each telescope requires 2 lens, 1 tube and 1 eyepiece
-HALF of the lenses in each pack are NOT useable (but we have to buy them anyway)
-We want to create telescopes with NO leftover parts (except for the lenses we can't use). [spoiler]
[/spoiler]
As mentioned at the beginning, this is an issue of MULTIPLES:
We need a number of telescopes that is a MULTIPLE of 25 (since we can only use half the lenses in each pack), 10 and 30. The least multiple is 300, BUT we need 300 lenses that we can USE, so we have to account for the extra lenses in each pack that we CAN'T use.
300 x 2 = 600
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
This question is perfect for TESTing THE ANSWERS (as Mitch has shown in his solution). It can also be solved with "math" by focusing on MULTIPLES of the numbers involved.
We're told that we can buy:
-Lenses in packs of 50
-Tubes in packs of 10
-Eyepieces in packs of 30
There are some additional details to consider:
-Each telescope requires 2 lens, 1 tube and 1 eyepiece
-HALF of the lenses in each pack are NOT useable (but we have to buy them anyway)
-We want to create telescopes with NO leftover parts (except for the lenses we can't use). [spoiler]
[/spoiler]
As mentioned at the beginning, this is an issue of MULTIPLES:
We need a number of telescopes that is a MULTIPLE of 25 (since we can only use half the lenses in each pack), 10 and 30. The least multiple is 300, BUT we need 300 lenses that we can USE, so we have to account for the extra lenses in each pack that we CAN'T use.
300 x 2 = 600
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
