Story Problem, not sure where to begin.

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Bpatters

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This is an extra credit problem where we are encouraged to ask family and friends to help. My family has no idea what to do and my friends aren't much help either.

Story:

A knight is completing quests for his king in order to finally retire. There are two different quests and each quest rewards gems. When completing the first quest the knight can choose either 60 emeralds, 25 rubies or 5 diamonds. When completing the second quest he can choose either 75 emeralds, 32 rubies or 7 diamonds. Each day the knight can only complete 4 of the first quest and 8 of the second quest. The knight needs 4,235 diamonds, 10,700 rubies and 43,900 emeralds to retire. How many of each rewards does he choose in order to retire the fastest.

I'm not sure where to begin on this one. I have tried to do some ratios to figure out which is better to choose each quest but that does not seem to work and there are so many variables I'm not sure which part I am supposed to be doing first. Any help would be much appreciated.

We need to know how may days it will take for the knight to retire and how many of each reward (4 rewards of quest one a day and 8 rewards of quest 2 a day) we need to choose total during the time it will take to retire.
 
Just found out where this problem came from... kind of funny

So one of the other students asked the teacher where this problem came from and it turns out the teacher plays a video game and this problem came up "IRL" for him so he did the math to figure out the quickest way to earn "his retirement" in his game and thought it would be a good exercise for us to figure out...
 
Got it so pretty much no matter what he will need to do those quests so many times in order to get enough... my only other thought is there a way to figure out if it would be better to say always take diamonds on the quest one because the ratio is higher than taking the rubies or emeralds... or maybe its pointless to hand in the quest 2 for diamonds because you get so few more than quest one compared to the others?
 
teacher says that there is a better answer :-/

it would be all of those combined because he can only get credit for one type of gem a day doing it that way not all of them... and i tried that answer but it is wrong because there is a way to figure out "the quickest" way for the knight to retire...

My teacher eluded to the fact that there is a way to figure out (because of the difference in numbers of gems from handing in quests one and two) that will save the night some time over all and noone in the class has gotten it right yet. So I'm still wondering how to go about figuring out (from the ratios?) if it would be better to only hand in a certain type of quest for quest one because over time it would save you time... apparently this problem is more complicated to get the correct answer and "get the Knight to retire the quickest"

So

first quest : either 60 emeralds, 25 rubies or 5 diamonds.
second quest : either 75 emeralds, 32 rubies or 7 diamonds.
but you can only hand in half of the number of quest 1 as quest 2
Each day : 4 of the first quest and 8 of the second quest.
Required : 43,900 emeralds, 10,700 rubies and 4,235 diamonds.
How many of each in order to retire the fastest?

so it much be a big equation like (.5(60a) or .5(25b) or .5(5c)) + (75a or 32b or 7c) = 43,900a + 10,700b + 4,235c or something similar to this?

Not sure how do you express "or" when you are writing an equation?
 
Either.

Bpatters said:
it would be all of those combined because he can only get credit for one type of gem a day doing it that way not all of them... and i tried that answer but it is wrong because there is a way to figure out "the quickest" way for the knight to retire...

My teacher eluded to the fact that there is a way to figure out (because of the difference in numbers of gems from handing in quests one and two) that will save the night some time over all and noone in the class has gotten it right yet. So I'm still wondering how to go about figuring out (from the ratios?) if it would be better to only hand in a certain type of quest for quest one because over time it would save you time... apparently this problem is more complicated to get the correct answer and "get the Knight to retire the quickest"

So

first quest : either 60 emeralds, 25 rubies or 5 diamonds.
second quest : either 75 emeralds, 32 rubies or 7 diamonds.
but you can only hand in half of the number of quest 1 as quest 2
Each day : 4 of the first quest and 8 of the second quest.
Required : 43,900 emeralds, 10,700 rubies and 4,235 diamonds.
How many of each in order to retire the fastest?

so it much be a big equation like (.5(60a) or .5(25b) or .5(5c)) + (75a or 32b or 7c) = 43,900a + 10,700b + 4,235c or something similar to this?

Not sure how do you express "or" when you are writing an equation?
Click to expand...

I was wondering about the word "either". Strictly speaking I think the problem is written correctly but is easily misinterpreted, e.g. :

first quest : either 60 emeralds, 25 rubies or 5 diamonds.

means,

first quest : either (60 emeralds, 25 rubies) or 5 diamonds.

not,

first quest : either (60 emeralds), or (25 rubies) or (5 diamonds).

?
 
yeah i guess i forgot that other or or used the wrong comma or something

you can only pick one gem as the reward per quest

so yes i think i forgot the other or

but only one reward per each quest per day

i'm not that good at English either with all those extra commas and stuff
 
Start with what Denis had
52 days for all emeralds = 43680 emeralds
30 days for all rubies = 10680 rubies
55 days for all diamonds = 4180 diamonds
Can we do better if all we require is those number of emeralds, rubies, and diamonds? What follows is a sort of heuristic argument that I think might be put on a firm foundation:
Well if we subtract some number of days from any one of them and use it for the others (possibly only one other), we will have more than we need for those others one so we will need to subtract some (possibly different) number of days from the others. Since we are going to have to add those subtractions back in to get to the required number for each gem type, we will either add (and originally subtract) in a least common multiple fashion or we will end up with more days. Thus the 137 days is the best we can do to get to that point.

That leaves 220 emeralds, 20 rubies, and 55 diamonds
Play with it and we need two days to complete the remaining [will need everything on quest 2 for diamonds for 1 day and 3 on quest 4 will not quite do it].

So 139 days minimum? If that is true, is it a coincidence that the ceiling of (0.26 + 0.06 + 0.72 = 1.04) is 2?
 
Ishuda said:
Start with what Denis had
...
So 139 days minimum?
Click to expand...
Condense problem to two quests a day [the 4 quest goes to one and the 8 quest goes to another]. A cycle is: Start at day one, pick two gem types at random, one type for one quest and one for the other. Keep count of the gems received and repeat until we have enough of each gem. If the number of days required has decrease, print out total of quests [a type 1 quest and a type 2 quest for each of the gems] and total of gems for the number of quests. Repeat for 320000 cycles
Results are
Number of days 139
Quest 1 [collapsed to one quest a day]
__Quests Emerald = 51 Ruby = 49 Diamond = 39
__Gems Emerald = 12240 Ruby = 4900 Diamond = 780
Quest 2 [collapsed to one quest a day]
__Quests Emerald = 53 Ruby = 23 Diamond = 63
__Gems Emerald = 31800 Ruby = 5888 Diamond = 3528
Total Gems Emerald = 44040 Ruby = 10788 Diamond = 4308
Code: 
Cycle     0;  MinD   164                             NUMBER OF DAYS TAKEN
 Quest 1 Emerald =    60   Ruby =    56   Diamond =    48           QUESTS
             Emerald = 14400   Ruby =  5600   Diamond =   960       GEMS
 Quest 2 Emerald =    54   Ruby =    51   Diamond =    59            QUESTS
             Emerald = 32400   Ruby = 13056   Diamond =  3304      GEMS
Cycle     3;  MinD   159
 Quest 1 Emerald =    61   Ruby =    50   Diamond =    48
         Emerald = 14640   Ruby =  5000   Diamond =   960
 Quest 2 Emerald =    49   Ruby =    51   Diamond =    59
         Emerald = 29400   Ruby = 13056   Diamond =  3304
Cycle     8;  MinD   151
 Quest 1 Emerald =    49   Ruby =    53   Diamond =    49
         Emerald = 11760   Ruby =  5300   Diamond =   980
 Quest 2 Emerald =    58   Ruby =    34   Diamond =    59
         Emerald = 34800   Ruby =  8704   Diamond =  3304
Cycle   202;  MinD   146
 Quest 1 Emerald =    50   Ruby =    41   Diamond =    55
         Emerald = 12000   Ruby =  4100   Diamond =  1100
 Quest 2 Emerald =    54   Ruby =    35   Diamond =    57
         Emerald = 32400   Ruby =  8960   Diamond =  3192
Cycle  2460;  MinD   145
 Quest 1 Emerald =    53   Ruby =    41   Diamond =    51
         Emerald = 12720   Ruby =  4100   Diamond =  1020
 Quest 2 Emerald =    52   Ruby =    34   Diamond =    59
         Emerald = 31200   Ruby =  8704   Diamond =  3304
Cycle  2604;  MinD   142
 Quest 1 Emerald =    51   Ruby =    43   Diamond =    48
         Emerald = 12240   Ruby =  4300   Diamond =   960
 Quest 2 Emerald =    53   Ruby =    28   Diamond =    61
         Emerald = 31800   Ruby =  7168   Diamond =  3416
Cycle 16784;  MinD   142
 Quest 1 Emerald =    52   Ruby =    36   Diamond =    54
         Emerald = 12480   Ruby =  3600   Diamond =  1080
 Quest 2 Emerald =    55   Ruby =    30   Diamond =    57
         Emerald = 33000   Ruby =  7680   Diamond =  3192
Cycle 20882;  MinD   142
 Quest 1 Emerald =    43   Ruby =    45   Diamond =    54
         Emerald = 10320   Ruby =  4500   Diamond =  1080
 Quest 2 Emerald =    57   Ruby =    28   Diamond =    57
         Emerald = 34200   Ruby =  7168   Diamond =  3192
Cycle 51700;  MinD   142
 Quest 1 Emerald =    42   Ruby =    39   Diamond =    61
         Emerald = 10080   Ruby =  3900   Diamond =  1220
 Quest 2 Emerald =    57   Ruby =    31   Diamond =    54
         Emerald = 34200   Ruby =  7936   Diamond =  3024
Cycle 62344;  MinD   142
 Quest 1 Emerald =    55   Ruby =    32   Diamond =    55
         Emerald = 13200   Ruby =  3200   Diamond =  1100
 Quest 2 Emerald =    53   Ruby =    32   Diamond =    57
         Emerald = 31800   Ruby =  8192   Diamond =  3192
Cycle 71633;  MinD   142
 Quest 1 Emerald =    48   Ruby =    32   Diamond =    62
         Emerald = 11520   Ruby =  3200   Diamond =  1240
 Quest 2 Emerald =    54   Ruby =    34   Diamond =    54
         Emerald = 32400   Ruby =  8704   Diamond =  3024
Cycle 88688;  MinD   141
 Quest 1 Emerald =    56   Ruby =    42   Diamond =    43
         Emerald = 13440   Ruby =  4200   Diamond =   860
 Quest 2 Emerald =    51   Ruby =    29   Diamond =    61
         Emerald = 30600   Ruby =  7424   Diamond =  3416
Cycle 98246;  MinD   141
 Quest 1 Emerald =    46   Ruby =    37   Diamond =    58
         Emerald = 11040   Ruby =  3700   Diamond =  1160
 Quest 2 Emerald =    56   Ruby =    30   Diamond =    55
         Emerald = 33600   Ruby =  7680   Diamond =  3080
Cycle 113729;  MinD   139
 Quest 1 Emerald =    51   Ruby =    49   Diamond =    39
         Emerald = 12240   Ruby =  4900   Diamond =   780
 Quest 2 Emerald =    53   Ruby =    23   Diamond =    63
         Emerald = 31800   Ruby =  5888   Diamond =  3528
 
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