Tom has 45 bushels of wheat, 63 bushels of rice, and 18 bushels of quinoa. He wants to order sacks of a single size such that he can fill each sack completely full of one type of grain without having any extra grain left over. What is the minimum number of sacks that Tom must order?
A) 9
B) 14
C) 18
D) 21
E) 42
OA = B
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Word problem - Grain sacks ordering
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adthedaddy
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GMATGuruNY
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Let x = the number of bushels that each sack can hold.adthedaddy wrote:Tom has 45 bushels of wheat, 63 bushels of rice, and 18 bushels of quinoa. He wants to order sacks of a single size such that he can fill each sack completely full of one type of grain without having any extra grain left over. What is the minimum number of sacks that Tom must order?
A) 9
B) 14
C) 18
D) 21
E) 42
Since no bushels are to be left over, the value of x must divide evenly into 45, 63 and 18.
To MINIMIZE the required number of sacks, we must MAXIMIZE how many bushels each sack can hold.
In other words, we must maximize the value of x.
45 = 3*3*5.
63 = 3*3*7.
18 = 3*3*2.
The values in red indicate that the greatest possible value of x = 3*3 = 9.
If x=9, then the values in blue represent how many sacks will be required to hold each type of grain:
5 sacks, each holding 9 bushels of wheat, for a total of 45 bushels of wheat.
7 sacks, each holding 9 bushels of rice, for a total of 63 bushels of rice.
2 sacks, each holding 9 bushels of quinoa, for a total of 18 bushels of quinoa.
Thus, the minimum number of sacks = 5+7+2 = 14.
The correct answer is B.
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Matt@VeritasPrep
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This question is asking (in a clever way) for the GREATEST COMMON FACTOR of 45, 63, and 18. Whatever that factor is will be the size of the sack.
GCF(18, 45, 63) = 9, so that's our size.
We then have to split the grain up into sacks of size 9, so we need 45/9 + 63/9 + 18/9 sacks, or 14 sacks, and we're done.
GCF(18, 45, 63) = 9, so that's our size.
We then have to split the grain up into sacks of size 9, so we need 45/9 + 63/9 + 18/9 sacks, or 14 sacks, and we're done.
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[email protected]
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Hi adthedaddy,
In this 'word problem', to answer this question in an efficient way, you really have to note the details about the sacks...
1) The sacks must be a SINGLE SIZE (so each sack will be the same size).
2) Each sack must be COMPLETELY FILLED with JUST ONE TYPE of grain (so we can't mix grains and we can't have a 'fraction' of a sack of grain).
Once you note these 'restrictions', the math itself is pretty straight-forward....
With 45 bushels of wheat, we can have any of the following options:
1 sack of 45 bushels each
3 sacks of 15 bushels each
5 sacks of 9 bushels each
9 sacks of 5 bushels each
15 sacks of 3 bushels each
45 sacks of 1 bushel each
The same can be done with the 63 bushels or rice and the 18 bushels of quinoa (although you don't really need to write down all of the options - you just have to see if any of the options that 'fit' the 45 bushels of wheat will also fit the 63 bushels of rice AND the 18 bushels of quinoa). From there, you should note that each of those numbers is ALSO divisible by 9.
Since the question asks for the MINIMUM number of sacks, we have to maximize the SIZE of each sack. A 9-bushel sack is the biggest one that 'fits' all three numbers, so we have...
45/9 = 5 sacks of wheat
63/9 = 7 sacks of rice
18/9 = 2 sacks of quinoa
5+7+2 = 14 total sacks
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
In this 'word problem', to answer this question in an efficient way, you really have to note the details about the sacks...
1) The sacks must be a SINGLE SIZE (so each sack will be the same size).
2) Each sack must be COMPLETELY FILLED with JUST ONE TYPE of grain (so we can't mix grains and we can't have a 'fraction' of a sack of grain).
Once you note these 'restrictions', the math itself is pretty straight-forward....
With 45 bushels of wheat, we can have any of the following options:
1 sack of 45 bushels each
3 sacks of 15 bushels each
5 sacks of 9 bushels each
9 sacks of 5 bushels each
15 sacks of 3 bushels each
45 sacks of 1 bushel each
The same can be done with the 63 bushels or rice and the 18 bushels of quinoa (although you don't really need to write down all of the options - you just have to see if any of the options that 'fit' the 45 bushels of wheat will also fit the 63 bushels of rice AND the 18 bushels of quinoa). From there, you should note that each of those numbers is ALSO divisible by 9.
Since the question asks for the MINIMUM number of sacks, we have to maximize the SIZE of each sack. A 9-bushel sack is the biggest one that 'fits' all three numbers, so we have...
45/9 = 5 sacks of wheat
63/9 = 7 sacks of rice
18/9 = 2 sacks of quinoa
5+7+2 = 14 total sacks
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
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sandipgumtya
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Hi Rich,
The language of the problem is still not clear to me.Can u pl explain once more what the problem is actually asking.
Thanks
The language of the problem is still not clear to me.Can u pl explain once more what the problem is actually asking.
Thanks
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Hi sandipbumtya,
In real basic terms, we need to pour 3 different amounts of grain into a certain number of sacks so that the following conditions are met:
1) Each sack MUST be the same size.
2) Each sack MUST be completely filled with JUST 1 type of grain.
3) We want the MINIMUM (re: least) number of sacks possible.
That third point is the 'key' - to have the least number of sacks, we need each sack to be as BIG as possible.
GMAT assassins aren't born, they're made,
Rich
In real basic terms, we need to pour 3 different amounts of grain into a certain number of sacks so that the following conditions are met:
1) Each sack MUST be the same size.
2) Each sack MUST be completely filled with JUST 1 type of grain.
3) We want the MINIMUM (re: least) number of sacks possible.
That third point is the 'key' - to have the least number of sacks, we need each sack to be as BIG as possible.
GMAT assassins aren't born, they're made,
Rich
