A cooperative farm has three subdivisions, A, B, and C. The subdivisions loan equipment to each other as needed. In the beginning, A loaned B and C as many reapers as each (meaning B and C) then had. Several months later B loaned A and C as many reapers as each (meaning A and C) then had. The following spring C loaned A and B as many reapers as each (meaning A and B) then had. Each subdivision then had 16 reapers. How many reapers did each have to begin with?
16 comments:
i dont get what it means that they loaned as many reapers as each then had?
For example if you and your friend both had three, and I loaned you as many reapers as you both had, I would loan you three as well as loaning three to your friend. So I would have to have 6 or more so that I could loan each of you three.
Hope that helps!
Well there has to be 48 reapers in all for all of them to have 16 at the end...
you can go in any pattern and end up with 16 each. If 'a' starts at 24 and give each of them 6 then 'a' has 12 and 'b' and 'c' both have 18. Then you do the same thing 'b' give each one 2 so 'b' has 14 'a' has 14 and 'c' has 20. Then 'c' give 'b' and 'a' each 2 and they all end up with 16. ('b' and 'c' start at 12 and 'a' starts at 24)
reapers. what are reapers? and can you explain it more.
There is a picture of a reaper with the problem. It is a machine for cutting standing grain; reaping machine.
Andrew- That is very perceptive. So there are 48 in all and must always be 48 in all every time there is an exchange.
You have to work backwards.
In the end, each subdivision had 16 reapers, a total of 48. Working backwards, C had given A and B the same amount of the reapers that they had. This means that the amount A and B had in the end has to be divided by 2 for each one, to get what they had before C loaned them. This means that A and B both had 8. Since, the total amount of reapers was 48, 48 has to subtract 16(8+8). So, C had 32 before lending A and B reapers.
Before that, B loaned A and C reapers which means that we must divide what A and C had before C loaned reapers by 2. This means that C had 16 reapers before B loaned reapers and A had 4 before B loaned reapers. 48-20=28 (20=16+4). So, before B loaned reapers, he had 28 reapers.
Iris,
You have excellent ideas! But remember... The idea of the blog is for you (and others) to give each other ideas on how to solve the problem (which you did). You suggested working backwards. Then you made an excellent suggestion about how to work backwards in THIS problem with the numbers in the problem. The Blog is not the place to explain the whole solution so only part of your explanation appears here. When you turn in the problem of the week (in next weeks problem) THAT IS WHEN you will go through a thorough explanation of the strategy and of how you arrived at your solution.
you have to find a way to make C have 32, B have 8, and A have 8 when c gives reapers
then when B gives a has to have 4, B has to have 28, and C has to have 16. so then at the beginning, A has to have 26, B has to have 14, and C has to have 8
It is true that this is an excellent problem in which to use the strategy "working backwards". If each had 16 reapers in the end, then there were 48 reapers in all. Prior to the last step if they all ended with 16, then A and B had to have less than 16. And if they were loaned as many as they had and then increased their amount to 16, then A and B each had 8, increased by 8 to get 16. So C had to have had enough to give A and B 8 each (16 in all). So C had 32, A had 8 and C loaned him 8 and B had 8 and C loaned him 8 too. So A=8 B=8 and C=32.
Then in the step before that B had to have enough to loan A and C. So if A had 8 after B loaned if as many as he had before, he must have had 4 and then B loaned him 4 to get 8. C has 32 after B loaned him so he must have had 16. And B had to have enough to give C 16 and A 4 so he loaned 20 and he must have had 28. Then in the first step, A loaned B and B had 28 so A must have loaned him 14. C had 16 after A loaned him so C must have started with 8 and A then started with 26.
NOW, to check (or justify) start with A=26, B=14 and C=8 and check each step going forward.(and you'll see it works!)
Well I know this is a little late but we have to work backwards and notice that they all loaned each other once.
A=8+8=16
B=8+8=16
C=16
I believe on our POTW 7
we should also use " working backwards"
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