Brain Teasers
Madadian Farm Animals
Old Farmer McDoughnut owns a piece of grassland in the heart of Madadia and has three animals: a Madadian Burger Cow, a Spare Kebab Goat, and a Scotch Egg laying Goose. McDoughnut discovered the following:
When the cow and the goat graze on the field together, there is no more grass after 45 days.
When the cow and the goose graze on the field together, there is no more grass after 60 days.
When the cow grazes on the field alone, there is no more grass after 90 days.
When the goat and the goose graze on the field together, there is no more grass after 90 days also.
For how long can the three animals graze on the field together?
When the cow and the goat graze on the field together, there is no more grass after 45 days.
When the cow and the goose graze on the field together, there is no more grass after 60 days.
When the cow grazes on the field alone, there is no more grass after 90 days.
When the goat and the goose graze on the field together, there is no more grass after 90 days also.
For how long can the three animals graze on the field together?
Answer
The cow, the goat, and the goose eat grass with a constant speed (amount per day): v1 for the cow, v2 for the goat, v3 for the goose.The grass grows with a constant amount per day (k).
The amount of grass at the beginning is h.
There is given:
When the cow and the goat graze on the field together, there is no grass left after 45 days. Therefore, h-45x(v1+v2-k) = 0, so v1+v2-k = h/45 = 4xh/180.
When the cow and the goose graze on the field together, there is no grass left after 60 days. Therefore, h-60x(v1+v3-k) = 0, so v1+v3-k = h/60 = 3xh/180.
When the cow grazes on the field alone, there is no grass left after 90 days. Therefore, h-90x(v1-k) = 0, so v1-k = h/90 = 2xh/180.
When the goat and the goose graze on the field together, there is also no grass left after 90 days. Therefore, h-90x(v2+v3-k) = 0, so v2+v3-k = h/90 = 2xh/180.
From this follows:
v1 = 3 x h/180,
v2 = 2 x h/180,
v3 = 1 x h/180,
k = 1 x h/180.
Then holds for the time t that the three animals can graze together: h-tx(v1+v2+v3-k) = 0, so t = h/(v1+v2+v3-k) = h/(3xh/180+2xh/180+1xh/180-1xh/180) = 36. The three animals can graze together for 36 days.
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Comments
Nice one - very sneaky / clever / sharp to add in that the grass is growing the whole time! I'd completely overlooked including that.
The model for grass growth is probably wrong! I don't know much about grass growth rates, but I guess they are partly dependant on how much grass is present (most population growth rates do). I realised the problem couldn't be solved ignoring grass growth, but felt that if included you should have ended up with differential equations, which seems a bit involved for a teaser!
A very good Maths problem (for you Mad). Thank you for the full solution. But I went to Madadia High School so I'm no good at Algebra. Here is how I figure it.
Consider what they eat in 180 days (the least common multiple of 45, 60 and 90).
(1)Cow + goat eat 4 fields in 180 days.
(2)Cow + goose eat 3 fields in 180 days.
(3) Goat + goose eat 2 fields in 180 days.
(4) Cow eats 2 fields in 180 days.
Subtracting (4) from (2) we find a goose eats 1 field in 180 days.
Subtracting this from (3) we find a goat eats 1 field in 180 days. Putting them together we get Cow + Goat + Goose eat 4 fields in 180 days.
BUT, we already knew that a Cow and a Goat alone eat 4 fieldS from (1) so the extra 1 field for the goose must come from the grass growing! Therefore there is the equivalent of 5 fields to be eaten in 180 days. ie The 3 animals can eat 1 field in 36 days. Nice puzzle
Consider what they eat in 180 days (the least common multiple of 45, 60 and 90).
(1)Cow + goat eat 4 fields in 180 days.
(2)Cow + goose eat 3 fields in 180 days.
(3) Goat + goose eat 2 fields in 180 days.
(4) Cow eats 2 fields in 180 days.
Subtracting (4) from (2) we find a goose eats 1 field in 180 days.
Subtracting this from (3) we find a goat eats 1 field in 180 days. Putting them together we get Cow + Goat + Goose eat 4 fields in 180 days.
BUT, we already knew that a Cow and a Goat alone eat 4 fieldS from (1) so the extra 1 field for the goose must come from the grass growing! Therefore there is the equivalent of 5 fields to be eaten in 180 days. ie The 3 animals can eat 1 field in 36 days. Nice puzzle
Nice Job
I worked this one out in my head like this:
From the last two equations it's easy to see the the cow eats grass at the same rate as the goat and goose together. By adding these two equations and subtracting two times the first equation, you can see that the goose eats grass at the same rate the grass grows. Finally, cancelling out the grass growing in the second equation (by removing the goose) and subtracting from the third shows that the cow eats grass at three times the rate it grows.
So the effect of adding the goose to the first equation is to cancel out the grass growing. Since the cow eats at three times the rate the grass grows and the goose eats at the same rate the grass grows, then the goat must eat at twice the rate of the grass, and the cow and goat together eat at five times the rate the grass grows. Therefore, if the goose cancels out the growing grass in the first equation then the cow and the goat will eat at 5/4 the rate they eat when the grass grows, finishing in 4/5 of the time.
(4/5) * 45 = 36
The explanation of the logic is much longer than what it took to reduce the problem this way. It was maybe thirty seconds to make all the connections and produce the answer without paper or pencil.
From the last two equations it's easy to see the the cow eats grass at the same rate as the goat and goose together. By adding these two equations and subtracting two times the first equation, you can see that the goose eats grass at the same rate the grass grows. Finally, cancelling out the grass growing in the second equation (by removing the goose) and subtracting from the third shows that the cow eats grass at three times the rate it grows.
So the effect of adding the goose to the first equation is to cancel out the grass growing. Since the cow eats at three times the rate the grass grows and the goose eats at the same rate the grass grows, then the goat must eat at twice the rate of the grass, and the cow and goat together eat at five times the rate the grass grows. Therefore, if the goose cancels out the growing grass in the first equation then the cow and the goat will eat at 5/4 the rate they eat when the grass grows, finishing in 4/5 of the time.
(4/5) * 45 = 36
The explanation of the logic is much longer than what it took to reduce the problem this way. It was maybe thirty seconds to make all the connections and produce the answer without paper or pencil.
When the cow grazes alone, it's 90 days.
When the cow and goat graze together, it's 45 days. Given how the cow eats 100% of it in 90 days, it eats 50% in 45 days. That means the goat eats 50% too. Therefore, the goat and cow eat at equal speeds with both requiring 90 days by themselves.
When the cow and goose graze, it takes 60 days. Given how the cow eats 100% in 90 days, it eats 2/3 in 60 days. That means the goose eats 1/3 in 60 days, which equals 180 days by itself.
All 3 of them together, in 180 days the cow grazes two, the goat grazes two, and the goose grazes one. 180 days/5 = 36 days.
When the cow and goat graze together, it's 45 days. Given how the cow eats 100% of it in 90 days, it eats 50% in 45 days. That means the goat eats 50% too. Therefore, the goat and cow eat at equal speeds with both requiring 90 days by themselves.
When the cow and goose graze, it takes 60 days. Given how the cow eats 100% in 90 days, it eats 2/3 in 60 days. That means the goose eats 1/3 in 60 days, which equals 180 days by itself.
All 3 of them together, in 180 days the cow grazes two, the goat grazes two, and the goose grazes one. 180 days/5 = 36 days.
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