How should four loaves of bread be divided into seven portions?

I would divide each loaf into two pieces, which gives eight. There would be one piece left over.

Crap. I just got to the end of the Details.
I was thinking of a nice cranberry stuffing.

Before I even loaded the page and read the details, I had already gone, ” 4 / 7 = 0.571… per person” and progressed to ”... half a loaf plus what would make 57%?” by the time the page loaded and I got to the last sentence of your first paragraph.

Given that most loaves have more than 14 slices, that “one-seventh of one-half” will still be enough for a sandwich or as a side for a single saucy entree or bowl of soup, while the intact half loaf can be cut (or not) as desired, leaving the recipient with more options. That makes the “half loaf plus a small hunk” a better way to do it than distributing 28 slices.

Anyways, as to the educational benefits of reading Count like an Egyptian, I have mixed feelings. My take on it is that people tend to absorb information that holds their interest more readily than they absorb things that bore them to tears, and learning math bores most people to tears.

Therefore, it stands to reason that most people don’t absorb much information about how to do math except that which they see as relevant to their daily lives. Most machinists aren’t geniuses, but they desire money enough to want to learn the math skills required to land the sort of paychecks you just can’t get flipping burgers. The same goes for carpenters, mechanics, and those who have technical hobbies. They all see a huge direct benefit in their lives, whether in income or enjoyment of a hobby, that makes them want to learn.

So while I feel it’s great to promote a book that is educational and illustrates alternative thinking, I also feel a strong “You can lead a horse…” vibe as I think that the actual shortcomings of our mathematical education are the result of a lack of thirst rather than a lack of water.

I agree with jca. That’s the only solution I can come up with without sitting down and thinking long and hard about it. Math is not my strong point, nor something I enjoy doing lol.

@OriginalCunningFox Well, @jca was off to a good start, but didn’t carry the remainder.

Think of it like this; everyone already has a half-loaf, but there is still half a loaf remaining. What would you do?

A) Throw that “extra” half a loaf away,
B) Distribute unequal shares by giving someone an extra half-loaf, or
C) Slice that remaining half-loaf up into seven equal pieces and hand them out as well
D) Rack your brains trying to figure out a more complicated solution and lose a few hours sleep over half a loaf of imaginary bread

Math kind of follows kid logic here. See, when kids pool their candy to share, they don’t throw any away, unfair trades cause more trouble than they’re worth, and they just don’t have the attention span to get terribly deep about calculating the “Optimum Fairness Equation” down to many decimal places.

Therefore, the average child will automatically eliminate A, B, and D from the list, or maybe never even see them as options, without even knowing any math at all! They’ll go for the simplest solution that seems fair without getting too deep into numbers (aside from counting), and if they can’t come up with something that can be split evenly, they carry the remainder over to the next exchange and cash in the “You owe me!” card.

In any event, it’s possible to solve “complex” math problems without using much math. Does that help any?

@jerv, Your solution is what I was looking for. I agree with you as to it being a natural solution that kids in a playground would come up with. And this is the argument made in the book, that the Egyptian method of doing arithmetic is more natural and easier than the way that we do it. Everything is presented in small understandable chunks that do not require any advanced math skills. It is accompanied by some information about what Egyptian life was like.

I don’t want to sound like a publicist for the book, but I like its pedagogical approach. It teaches math in a historical context. It presents an alternative way of thinking about arithmetic that illustrates the fact that there can be more than one way of solving a problem. The author makes a good case for Egyptian arithmetic as being easier to learn and use, though I would not suggest abandoning our way.

It would be interesting to see if there are benefits to first introducing elementary school students to the Egyptian method before introducing the standard way. The advantage is that students could learn it much faster than the standard method and would later be able to use it to check their answers when they switch over.

It depends how precise you really need to be.

1. I would likely cut the loaves 60/40, or whatever the precise cut is (let’s use 60/40 for simplicity) and four people get 1/7th in one piece and one of the smaller “halves” be cut in thirds to finish the problem. Does that make sense?

2. Or, cut all loaves in half and give out 7 with the one remaining to be saved for later.

3. Or, cut each loaf in half and the last quarter into seven pieces to give out evenly, this only works with a large loaf of bread in my opinion.

I like number 1 best.

Edit: to be precise with my #1 method, I would measure the loaf in centimeters, multiply by 8, divide by 7, and then measure where to cut the loaves in “half” using that answer.

Hmmm…WWJD?

Give 3.90 loaves to hedge fund mangers?

My solution reminds me of when we get pizza for a meeting. There’s usually a leftover slice or two, but not a big tragedy if one person gets it and the others don’t. How would one slice a loaf of bread (or pizza or whatever) to make such a precise measurement such as suggested above? It’ might be accurate but it’s not practical.

Correction: It should be multiply by 4 divided by 7. I wrote my answer around 4:am. Not my best time of day.

@jca If it doesn’t need to be exist it’s way easier. Just sort of guesstimate it, and some get more and some get less. Some people might want the smaller ones and some might want the larger. If it’s a give away no problem. If you’re paying for it then people might be a little more picky.

Give everyone a half loaf.

Whoever is still hungry gets the last half.

Who’s bringing the butter?

Cut each loaf into 7 equal slices and give everyone 4 slices.

Pass the butter, hoard the honey.

I am surprised that people are not finding as natural @jerv‘s extension of @jca‘s answer. Give everyone half a loaf and then divide the remaining half loaf into 7 parts. Dividing in half should be fairly easy and nobody is going to complain much if they get slightly less than 1/7 of the last half loaf.

Numerically this works out to 4/7 = ½ + 1/14, since 1/7 of ½ = 1/14. The Egyptian fractions had no numerators. The only idea they could express was that of dividing something into equal parts, what we would consider fractions of the form 1/n.

It is not difficult to prove algebraically that the Egyptian method will always work. I can show the proof for anyone who might be interested. Any fraction a/b can be expressed as a/b = 1/n1 + 1/n2 + ... + 1/nk for some finite number k, with the nk increasing in size, meaning that the 1/nk keep getting smaller.

For example, suppose the problem was to divide 4 loaves among 5 people. Divide the 4 loaves in half, giving 8 halves. Give each person a half, leaving 3 half loaves. Next divide the 3 half loaves in half, giving 6 quarter loaves. Give each person one quarter loaf, leaving one quarter loaf, which can now be divided into 5, giving each person one part. 4/5 = ½ + ¼ + 1/(20). In the general case, it may be necessary to divide by some number other than 2 in order to cover the number of people, but this method will always work and eventually come to an end.

I picture a small loaf and dividing a half into 7 pieces as difficult. If it’s a big loaf then it’s the easiest solution. Plus, when thinking of a loaf, I don’t think of sliced bread, I think of cutting off a chunk. Are we making sandwiches? Or, are we eating the bread plain? With butter? Not that it matters much. One thin slice won’t be traumatic.

Like the three sons who inherited 17 camels. How to divide? One man came along and said I will lend you my camel. One son received ½ = 9 camels. One son received ⅓ = 6 camels. And the third son received 1/9 = 2 camels. 9 + 6 + 2 = 17. And the man walked away with his “loaner” camel.

@jerv Wow, I learned some things about math with your solution there haha. Thanks for the explanation!

The question wasn’t about dividing the loaves into equal portions. It just said “seven portions.” With dividing it into 8 (each loaf divided into two), then it’s easy math, quick cutting, less cuts, and one piece left over for either one person to take it, or whatever.

@jca I agree with that method if it’s simply for 7 people and no need to try to divide the 4 loaves perfectly equal. All seven people will get equal portions and you can take the extra for doing the cutting.

@JLeslie Most loaves I’ve seen have more than 14 slices to them, so I figured that the seven slices you’d get from a half-loaf would each be at least enough for a big bowl of French onion soup. I’m thinking about “Texas Toast” thickness; hefty, but not quite enough to slice again to make a sandwich.

@LostInParadise I agree, but my knowledge of human nature (how kids learn, the institutional inertia of the education system, etcetera) makes me have reservations.

@jca Yes, the “equal” was implied rather than stated. However, unequal solutions like slicing one loaf into six pieces (which can be done in three cuts), flipping the other six people the bird and walking away with three whole loaves, while semantically valid, explains why mathematicians and logicians don’t have many friends. Still, good point.

“Most loaves I’ve seen have more than 14 slices to them.” @jerv

How is slicing 4 loaves into 7 slices less efficient?

@ibstubro As most people cut loaves with a knife rather than a multi-bladed machine, making 28 slices will involve 24 cuts; six cuts per loaf.

4 * 6 = 24

Slicing loaves in half takes four cuts, and slicing a half-loaf into seven slices is six more cuts.

4 + 6 = 10

So let’s compare…

10 < 24

Now, if you are the type of person who uses a sword instead of a knife in the kitchen and has extra hands to hold all four loaves still at once, I could see the 28-slice solution being more efficient. Likewise, if I had one of those big slicers like the supermarket bakery next door has, well, I could just load all four loaves in the feed hopper and get the job completely done in one pass.

However, I don’t have one of those machines, my kitchen has no blades longer than 10”, and I only have two hands. Since one of my hands would be holding the knife, I’d only be able to securely cut one loaf at a time. While I could cut one loaf lengthwise, there is no way I could secure two or more at once in any orientation unless we were talking about a baguette that had no business being in my kitchen to begin with. (If I wanted that much crust, I’d eat burnt crackers; they’re nothing but crust!)

Once again, non-mathematical knowledge and a little common sense helps solve complex math. In this case, the only ways to get 28 pieces involves either equipment or extremities that I don’t have or a larger number of cuts.

The “half-loaf plus slice” solution starts out with four cuts handling the majority of the problem, and figuring out out how do split one half-loaf seven ways (1/7) is a FAR simpler problem than the original problem (4/7) that can be handled by doing once to the half-loaf what you wanted to do to all four whole loaves.

Doing something once is more efficient than doing it four times, and even adding the four cuts it took to get to the intermediate step of eight half-loaves in the first place, well, those four cuts eliminated the need for 18 cuts (what it’d take to cut 3 loaves into 21 slices), so you’re still ahead by 14 cuts.

Does it make more sense now?

@jerv I’m thinking homemade bread, since it’s something being given away, and it could be any size. I’m just telling you what I pictured in my head; a small loaf. Jews don’t usually jump to sandwich bread I don’t think when we hear bread. I know I don’t. I’m in the French baguette, a rustic round, a rye loaf, maybe even banana or zucchini bread. My husband sometimes prefers I don’t get the loaf sliced if I buy it at the store, and my mom is the same. But, I do agree most of the time you probably could divide one side into seven pieces. Seven thin slices by hand sounds tedious and not worth the time.

More and more I like just cutting all the loaves in half and then the eight piece can be given to people who want some extra (it could be divided between two people easily) or put in the freezer for later, or extra for the host.

@JLeslie Ah. See, most of the Rustic Rounds I’ve seen have been over 10 inches across, and my wife tends to make fairly large loaves when she makes bread, so I tend to think of those sort of loaves in addition to pre-sliced sandwich loaves. But it seems that all the different loaves I thought of were still rather larger than what you pictured when you thought “loaf”.

Since I tend to use pieces considerably thinner than a full-inch (unlike snakes, I cannot unhinge my jaw), I didn’t think it tedious to cut a half-loaf into seven slices. But it would seem that my half-loaves are about the size of your full loaves, and I can see how it would be problematic with the smaller loaves you’re used to.

With a round loaf, dividing it equally when using straight slices would be very difficult, because the ends are smaller then the middle. Unless, of course, you use my method of halving the loaves, or even halving the halves, then it’s equal. Regular slices, as we know when slicing homemade bread, is tough with the ends. So I go back to my method – 8 half loaf pieces with one left over.

@jca Did wedges not occur to you? While not exactly good for sandwiches, they work perfectly fine for grabbing sauce off a plate.

Now, your solution may work if you were the one that made/bought the bread. In that case, nobody would fault you for keeping a whole loaf for yourself and they would just be grateful for the gift of a half-loaf.

If, on the other hand, you and six of your friends had four loaves to split between the seven of you, you would either come up with a better way than you have already regardless of the shape or size of the loaves or you would risk starting a fight. I’ve seen enough strife caused by “one for you, two for me” unequal sharing that I consider fairness worth being more precise than +/- 15% even if it is a little extra work, especially if it’s an almost trivial added effort.

It seems that you and I have VERY different views on how to handle fractions. Far more different than just being used to smaller loaves than me like @JLeslie is. And I’m having a hard time wrapping my head around how that degree of imprecision is even acceptable. If that’s good enough for you, c’est la vie, but I personally have never been any place where I could afford to be that sloppy or inequitable without major repercussions. It’s a foreign concept to me. Almost completely alien.

@jerv: It’s ok that you and I don’t necessarily agree on how to divide some loaves of bread. I feel, for myself, that enough time has been spent on this issue.

If this were a true life situation, nobody that I know or nobody that I care to hang out with would throw a hissy fit over not receiving a piece of bread. If they did, that would be the last time I spent time with them.

@jca I think so too.

Unfortunately, I’ve often been in situations where I was forced to deal with those prone to hissy fits, leaving me no choice but to deal with the aftermath whichever way they go; not hanging out with someone just isn’t an option sometimes.

@jerv: That is most unfortunate.

The bread’s stale by now. Should we make bread-pudding or dressing?

Dressing for me. Oyster or cranberry, please.