Aunt Hilda Problem

Definition of variables :

• m: the amount of snow which is falling per day on the area of the university
• t: the amount of days the teams are waiting before they start with shovelling snow
• a: amount of snow the first team (2 persons) shovelled
• b: amount of snow the second team shovelled (5 persons)
• c: amount of snow the third team shovelled (11 persons)
• p amount of snow one person shovels on one day
• x: number of days the third team needs to shovel the university snow free

Hence m*t is the amount of snow which has fallen before the teams start to shovel

Team A shovels 1/4 of the university snow free, which means they shovelled 1/4 of the amount of snow which has fallen before they started (m*t) and 1/4 of the amount of snow which has fallen during the shovelled, which is 8*m, because they shovelled 8 days.

expressed in a formula this is :
a = 1/4*(m*t) + 1/4*(8*m) = 1/4*(m*t + 8*m)

similar we gain for the other teams the formulas
b = 1/2*(m*t + 6*m) and c = (m*t + x*m)

According to the hint we now try to calculate the amount of snow one person shovels per day. This is the total amount of snow one team shovels divided by the number of days and the number of persons.

p = a / (2*8) = a / 16
p = b / (5*6) = b / 30
p = c / (11*x)

If we now equate the upper two equations and insert the expressions we have for a and b we have the following

(1/4(mt+8m))/16 = (1/2(mt+6m))/30
1/64(mt+8m) = 1/60(mt+6m)

now we have to multiply with the smallest common denominator, which is 15*16*4 = 960 and results in

15(mt+8m)=16(mt+6m)
mt = 24m
t = 24

Voila, we have our first variable determined. Now we use the lower two equations and do the same again

(1/2(mt+6m))/30 = (mt + xm)/11x

and insert the value of t, 24

1/60(24m+6m) = (24m + xm)/11x
m/2 = m*((24+x)/11x)

divide through m

1/2 = 24+x/11x
11x/2 = 24 + x
11x = 48 + 2x
x = 5 1/3

And we are finished the needed 5 and 1/3 days to shovel the hole university snow free. ( assumed that they shovelled all day and night :)