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Pills they had 2 (Posted on 2006-06-13) Difficulty: 5 of 5
Continuing on the pills problem.

Mowing a lawn 5 acres in size is part of a community service project. A group of people, including boys, girls, ladies and men, take up the job. They continue until the job is done. The grass also is growing at a constant rate hence the job increases if the number of days are higher but on the opening day job is the same.

a) 6 boys and 3 ladies can do the job in 4 days.
b) 8 girls & 3 men can do it in 3 days, which is the same as the time taken by 3 men and 6 Ladies.
c) Men work twice as fast as boys.
d) If only a girl were to do the job she would never be able to complete the job.

One day they reach the place of work and find that there are four bowls containing pills ( Either Poison pill or Energy pill ) kept for each category. (Marked- Boys, Girls, Ladies & Men). After consuming a poison pill their rate reduces to one third and after consuming an energy pill their rate becomes double. Three of these bowls contain one type of pill & one contains the other type.

In all, eight persons are present and there is at least one member from each category. They each take one pill from their respective bowl and start working. The job was completed in 9 days and the number of ladies is not more than the number of boys. How many boys, girls, ladies and men were present and which pill did they have?

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Solution Not a computer but a solution nonetheless (spoiler) Comment 2 of 2 |

Answer: 1 lady on energy pill, 1 man poisoned, 2 girls poisoned and 4 boys poisoned.
Start by writing the equations for a single days effort:

For ease of explanation I divided the lawn into 864 subplots (3^3 * 2^5)

T = Total area
d = day's growth

(1)  6b + 3L = T/4 + d
(2)  8g + 3m = T/3 + d
(3)  3m + 6L = T/3 + d
(4)  6b = 3m
(5)  g <= d

Substituting (4) into (3) and then subtracting (1)
3L = T/3 T/4
L = T/36
  L = 24 subplots per day
Subtracting (2) from (3) gives
  g = 18 subplots per day
Now assuming that d = g then d = 18 subplots per day worth of growth then it can be found that
  m = 54 subplots/day
  b = 27 subplots/day

To mow the lawn in 9 days, the group must mow 864/9 = 96 subplots plus the amount of growth = 18 subplots for a total of 114 subplots per day.

Lets add in the funny pills:

If a man takes an energy pill he could do 108 subplots per day.  This, with any cooperation from 7 other folks would complete the job in less than 9 days. So all the men take poison pills.

Since there is 1 of one type of pill and 3 of the other, assume the 3 are energy pills:

Then 1 boy, 1 girl and 1 lady would complete 2 * (18 + 24 +27) = 142 subplots/day. Too many subplots and thus there are not 3 energy pills there is only 1 energy pill.

Manipulating the placement of the pills, I found two solutions:
1 g (energy) and  3 L, 2 m, 2 b (poison)
1 L (energy), 2 g, 1 m, 4 b (poison)
Since the ladies do not outnumber the boys, only the second solution works.

1 lady with the blue pill will do 1 * 24 *2 subplots per day. (48)
2 girls with the red pill will do 2 * 18 / 3 subplots per day. (12)
1 man with the red pill will do 1 * 54 / 3 subplots per day. (18)
4 boys with the red pill will do 4 * 27 / 3 subplots per day. (36)
Total 114 subplots / day

Edited on June 13, 2006, 4:52 pm
  Posted by Leming on 2006-06-13 16:51:40

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