**1. A Clever Solution**

*(This comes from a friend who did not identify the source.)*

A farmer died, leaving his 17 horses to his three sons. When his sons opened up the will it read:

- My eldest son should get 1/2 (half) of total horses;
- My middle son should be given 1/3 (one third) of the total horses;
- My youngest son should be given 1/9 (one ninth) of the total horses.

As it’s impossible to divide 17 into halves, thirds or ninths (that is, divide 17 by 2, 3 or 9), the three sons started to fight with each other. Eventually they decided to go to a farmer friend whom they considered quite smart, to see if he could work it out for them. The farmer friend read the will patiently, and after giving it due thought, he brought one of his own horses over and added it to the 17. That increased the total to 18 horses.

Now, he divided the horses according to their father’s will.

- Half of 18 = 9. So he gave the eldest son 9 horses.
- 1/3rd of 18 = 6. So he gave the middle son 6 horses.
- 1/9th of 18 = 2. So he gave the youngest son 2 horses.

Now add up how many horses they have:

- Eldest son 9
- Middle son 6
- Youngest son 2

The TOTAL IS SEVENTEEN.

This leaves one horse, so the farmer friend takes his horse back to his farm.

Problem Solved!

Moral:

The attitude of negotiation and problem solving is to find the 18th horse i.e. the common ground. Once a person is able to find the 18th horse, the issue is resolved. It is difficult at times. However, to reach a solution, the first step is to believe that there is a solution. If we think that there is no solution, we won’t be able to reach any!

**2. Same Problem, Done The Common Core Way.**

*(This is my take on trying to find a common core compliant solution.)*

First and foremost, there is the issue of “social justice.”

1. Explain why the old farmer hated his middle and youngest sons so much.

2. Consider if it would be fair to divide the horses equally among the brothers.

3. To see what would be fair, consider if the brothers already have horses of their own, and how many.

4. if the oldest son already has many horses, is it fair that he would get most of his father’s horses, too?

5. Discuss the environmental impact of 17 horses, and how it relates to global warming.

But let’s say the old farmer had his reasons and they are good and fair to his three sons. So,

Half of 17 is….

Well, let’s try our friendly number 5.

- Five horses to the oldest son, that leaves….
- Add 5 to make 10, 5 more to make 15, 2 more to make 17.
- Now, 5+5+2=12; that’s too much more than 5. OK,…

Try our friendly number 10….

- Add 5 to make 15, add 2 more to make 17.
- 5+2=7, but now 10 is too much more than 7. Hmm…

Oh. Oh… I know. Let’s DIVIDE 17 by 2.

- Why, you ask? Because by definition half of something means dividing it into two parts, which means dividing it by 2.
- Using the only way we know to do division, which is repeated subtraction…,
- Why, you ask? Because by definition division is subtraction repeated till you have nothing left, which in this case means keep subtracting 2.
- 17-2=15; You can do this in your head, right?
- 15-2-13
- 13-2=11
- 11-2=9
- 9-2=7
- 7-2-5
- 5-2-3
- 3-2=1
- 1-2=… Nope, gone to far.

OK, we have subtracted 2 from 17 (count the steps, 1,2,3,4,5,6,7,8) 8 times so the answer is 8, with a remainder of 1.

You can also draw dots and draw circles around pairs of them, and count the pairs.

Or you can put coins on the table, keep pulling aside two coins at a time, and count the pairs.

Either way, Number One Son gets 8 horses.

Now let’s repeat the procedure for the middle son. He is supposed to get 1/3rd of the herd, so let’s divide 17 by 3.

- 17-3=14 You can do this in your head, too, right?
- 14-3=11
- 11-3=8
- 8-3=5
- 5-3=2
- 2-3=…

You can also draw dots and draw circles around any three of them, and count the trios.

Or you can put coins on the table, keep pulling aside three coins at a time, and count the trios.

Either way, Number Two Son gets 5 horses.

Now let’s repeat the procedure for the youngest son. He is supposed to get 1/9th of the herd, so let’s divide 17 by 9.

- 17-9=…
- Well, this time it’s best you use the number line. Draw a line, put tick marks in it from 1 to 17, and count 9 tick marks backwards from 17.
- 17,16,15,14,13,12,11,10, 9. OK, that was 9 tick marks, and it got us to 9 on the number line. That’s once. Continue…
- 9,8,7,6,5,4,3,2,1. OK, that’s another 9 tick marks, and it got us to 1 on the number line. That’s twice.
- There are no more tick marks left, so the answer is 2.

Number Three Son gets 2 horses.

Now check the answer.

- Number One Son gets 8 horses,
- Number Two Son gets 5 horses.
- Number Three Son gets 2 horses.
- 8+5+2= … (well, remember your friendly number 10)
- 8+2+5=
- 10+5=
- 15.

Oops. There are 2 horses left that did not go to either son. But who cares? **Common Core cares about the PROCESS, not whether it leads to the CORRECT ANSWER.**

So, the wise neighbor way, above, allocates 9+6+2 horses among the brothers and accounts for all 17 of them.

The Common Core way gives them 8+5+2. It accounts only for 15, leaving the brothers with the problem, how to divide the two remaining horses.

Who is to say which solution is correct?

- Maybe Number One Son will let each of his younger brothers have one extra horse, giving them 8, 6 and 3 respectively.
- Or, being the typical oldest brother, he’ll keep them both (10, 5, 2).
- Or they’ll sell two horses and divide the money instead. Then they can repeat this mess to figure out how much money each brother gets.

Isn’t Common Core / social justice math oh so wonderful?

Carole Fineberg

said:Excellent analogy.

kodabar

said:The difficulty is that the original problem is not solved by the 18th horse. It seems to be, but only if you forget part of the problem.

Forget the 17 horses for a moment. Just think about the fractions.

1 – (1/2 + 1/3 + 1/9) = 1/18

So there’s a one eighteenth share of the horses that is not given to any of the sons. That is either given to a fourth party not mentioned in the problem or goes to the state because it’s unaccounted for in the will.

When they borrow the 18th horse, they don’t just add that one in, they also add in the fourth party’s share. The 18th horse is irrelevant. They solve the problem through theft. Everyone ignores this bit.

The common core stuff is absolutely fine, though I’d add a question six: Who got the farm? (He was a farmer, remember?) Even if the farm was rented, there would still be farming equipment and machinery. And there would be animals and/or crops too.

peter5427

said:There was no theft. There was no fourth party. The neighbor simply took his own horse back. The rest of it (about the farm, etc.) is just noise, typical of the Common Core way of doing anything.

kodabar

said:Add it up. One brother gets a half of the horses. One brother gets a third. One gets a ninth. A ninth plus a third plus a half equals seventeen eighteenths. To who does the one eighteenth belong? This is before the neighbour’s horse is brought in.

When the neighbour’s horse is added, the outstanding one eighteenth is also added. It’s not the neighbour’s horse that enables them to resolve this, but the one eighteenth that wasn’t accounted for in the initial share.

Watch, I’ll do it in decimal maths:

There are 17 horses.

One son gets half the horses. That’s 8.5 horses.

One son gets a third of the horses. That’s 5.6666… horses.

One son gets a ninth of the horses. That’s 1.8888…. horses.

That’s a total of 16.0555… horses.

There’s 0.9444… horses unaccounted for. They still belong to the estate and are not divided amongst the sons. So the fourth party is the estate (or an unnamed person not mentioned in the story).

By adding the neighbour’s horse, it doesn’t help. It seems like it does because you then divide 18 horses evenly among the sons and have one left over. But really, you’re adding not only the neighbour’s horse, but the 0.9444… of a horse that was excluded from the original calculation.

Son 1 was getting 8.5 horses, but now he gets 9. That’s an increase of 0.5

Son 2 was getting 5.666… horses, but now he gets 6. That’s an increase of 0.333…

Son 3 was getting 1.888… horses, but now he gets 2. That’s an increase of 0.111…

And there’s a horse left over. How can they all get an increased share without any increase in the total number of horses?

0.5 + 0.333… + 0.111… = 0.9444…

The increase in their shares is entirely due to the 0.9444… of a horse that was unaccounted for in the original split. It has nothing to do with the neighbour’s horse. All that horse did was make the maths easier.

They have stolen the 0.9444… that was not assigned to them.

If you don’t agree, explain the error in my maths. Explain how a half plus a third plus a ninth = one. A half plus a third plus a SIXTH = one. The sons were left 16 and a bit horses, but have kept seventeen. They have indeed stolen most of a horse.

I think you’ve found a better solution through the common core approach. It provides an equitable and fair solution that doesn’t involve ownership of partial horses.

Compare this to the missing dollar riddle. It’s an informal fallacy where the presentation of the question confuses one into accepting something fallacious, even though all the steps are seemingly logical.

https://en.wikipedia.org/wiki/Missing_dollar_riddle

peter5427

said:I’ll give your analysis the serious consideration that it still does not deserve when you can show me how you can have half a horse or 88.8% of a horse — and still have a horse. We were NOT talking about butchering these poor animals. In the meantime, please consider that you CAN’T be stealing anything when you are INHERITING it.. or borrowing it to make the math easier..