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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 one solution; other possibilities; -- computer assisted (spoiler) | Comment 1 of 2

If l, g, b and m are the rates of cutting of ladies, girls, boys and men respectively, in fields per day, and r is the rate of grass growth in fields per day, then

6b + 3l - r = 1/4
8g + 3m - r = 1/3
3m + 6l - r = 1/3
m = 2b

also g-r <= 0

From these we get

l = 1/36
g = 1/48

r >= 1/48

b = 1/36 + r/6
m = 1/18 + r/3

The following program assumes r = 1/48 and tries all possible numbers of ladies, girls, boys and men that add up to 8, and all possibilities of what might be the sole poison bowl or the sole energy bowl:

DEFDBL A-Z

CLS
cat$(1) = "ladies"
cat$(2) = "girls"
cat$(3) = "boys"
cat$(4) = "men"
lRate = 1 / 36
gRate = 1 / 48

r = gRate
bRate = 1 / 36 + r / 6
mRate = 2 * bRate

FOR noL = 1 TO 5
FOR noG = 1 TO 6 - noL
FOR noB = 1 TO 7 - noL - noG
  noM = 8 - noL - noG - noB
  energy = 0
  FOR poison = 1 TO 4
   l = lRate
   g = gRate
   b = bRate
   m = mRate
   SELECT CASE poison
    CASE 1
     l = l / 3: g = g * 2: b = b * 2: m = m * 2
    CASE 2
     l = l * 2: g = g / 3: b = b * 2: m = m * 2
    CASE 3
     l = l * 2: g = g * 2: b = b / 3: m = m * 2
    CASE 4
     l = l * 2: g = g * 2: b = b * 2: m = m / 3
   END SELECT
   GOSUB eval
  NEXT
  poison = 0
  FOR energy = 1 TO 4
   l = lRate
   g = gRate
   b = bRate
   m = mRate
   SELECT CASE energy
    CASE 1
     l = l * 2: g = g / 3: b = b / 3: m = m / 3
    CASE 2
     l = l / 3: g = g * 2: b = b / 3: m = m / 3
    CASE 3
     l = l / 3: g = g / 3: b = b * 2: m = m / 3
    CASE 4
     l = l / 3: g = g / 3: b = b / 3: m = m * 2
   END SELECT
   GOSUB eval
  NEXT
NEXT
NEXT
NEXT
END
eval:
  tr = l * noL + g * noG + b * noB + m * noM - r
  IF ABS(tr - 1 / 9) <= .000000001# THEN
    FOR i = 1 TO 4: PRINT cat$(i); " "; : NEXT: PRINT
    PRINT noL; noG; noB; noM
    PRINT USING " #.#########"; lRate; gRate; bRate; mRate
    PRINT USING " #.#########"; l; g; b; m
    PRINT USING " #.#########"; r
  END IF
RETURN

The results remind us of the order and give lines that represent
how many are present
what their ordinary rates are (fieldsworth per day)
what their rates are after the pills

Also shown is the grass growth rate:

ladies girls boys men
 1  2  4  1
 0.027777778 0.020833333 0.031250000 0.062500000
 0.055555556 0.006944444 0.010416667 0.020833333
 0.020833333
ladies girls boys men
 3  1  2  2
 0.027777778 0.020833333 0.031250000 0.062500000
 0.009259259 0.041666667 0.010416667 0.020833333
 0.020833333
ladies girls boys men
 3  3  1  1
 0.027777778 0.020833333 0.031250000 0.062500000
 0.009259259 0.006944444 0.062500000 0.020833333
 0.020833333
 
The second and third solutions each have more ladies than boys, so the first solution is the one desired, with 1 lady, 2 girls, 4 boys and 1 man. Only the ladies have taken the energy pill and the others have taken poison pills.

But the above assumes that one girl is just able to keep up with the growing grass. Another possibility is r > g, where a girl by herself with no pills will lose ground to the growing grass.  Such a situation is this:

ladies girls boys men
 1  4  1  2
 0.027777778 0.020833333 0.034722222 0.069444444
 0.009259259 0.006944444 0.069444444 0.023148148
 0.041666667
ladies girls boys men
 2  1  2  3
 0.027777778 0.020833333 0.034722222 0.069444444
 0.009259259 0.041666667 0.011574074 0.023148148
 0.041666667
 
 In either of these scenarios the number of ladies equals the number of boys. The rate of grass growth is twice the cutting power of a girl. In the first only the boys have energy pills and the rest poison pills.  In the second, only the girls had energy pills and the rest poison.
 
 I'm sure by varying the grass growth rate, other scenarios could apply, but presumably the author intends that one girl exactly keeps up with the growing grass.


  Posted by Charlie on 2006-06-13 15:47:47
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