We received lots of good solutions to
this problem - well done everyone! Many of you spotted that Mr
McGregor should put 7 plants in his potting shed at the
beginning, and put 8 plants in each garden. Well done to Henry
from Finton House School, Ruth from Manchester High School for
Girls, Liam from Wilbarston School, Mel from Christ Church
Grammar School, Rachel from Beecroft Public School in Australia,
Yanqing from Devenport High School for Girls and Daniel from
Junction City High School for their detailed explanations of how
they arrived at the answer.
Henry from Finton House School
wrote:
"1 x 2 x 2 x 2 = 8.
8 therefore seems likely to be the number in the garden. Let's
try it.
The number in the shed at the end must be the number in the
garden.
Now what number do we double to get to 8? It must be 4.
4 + 8 = 12. 12 divided by 2 = 6.
6 + 8 = 14. 14 divided by 2 = 7."
Liam used similar logic:
"Just work backwards from the last garden. Imagine there to be 8
plants in each garden. (You can't have odd numbers in a garden as
the last garden must be double the whole number of plants left
after the 2nd garden was planted. I chose 8 because it's a
conveniently sized power of 2.) There must have been 4 plants
left after the 2nd garden was planted so before it was planted
there must have been 12 which is double 6. 6+8=14. So Mr McGregor
needs to put 14/2 or 7 plants in his magic potting shed at the
beginning!"
Yanqing and Rachel used algebra. Here is
Yanqing's solution:
"First, we make the number of plants put in the shed n,
and the number planted each night x.
So by the first morning, the number has doubled to 2n in the shed.
We plant x of them, leaving 2n-x in the shed overnight.
By the second morning, we have 2(2n-x)=4n-2x in the shed.
Planting x of them, we are left with 4n-2x-x=4n-3x in the shed.
By the third morning, there should be 2(4n-3x)=8n-6x plants in the shed.
There need to be x plants in the shed,
as we need to plant all of them, so 8n-6x=x and 8n=7x.
We can now say that the ratio of n to x is 7:8,
so the smallest values for n and x,
where they are both positive whole numbers, are obviously 7 and 8.
Other numbers which will work are all multiples of 7."
Rachel also found that 8n = 7x and concluded that:
"Now you can see that 8n or 7x could equal 56, which makes
n = 7 and x = 8.
This works when you try it out, and if you multiply both numbers
by another number, those new numbers work too."
Daniel concluded that:
"If you want to have the same amount of plants in each
garden you must start with a multiple of 7 plants in the shed and
each day plant the same multiple of 8 plants in the garden."
James from C.G.S.B found such a solution:
"Start with 35 ...then put 40 in each garden"
And so did Mel from Christ Church Grammar School!
"You start off with 301 plants in the shed. You put 344 in each
garden."