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Here is the question that confused me:

$$\text{What is the value of}\;\frac15\div\frac15\div\frac15\div\frac15\div\frac15\div5\div5\div5\;? \tag1$$

If the signs stay the same, is the operation done LTR or RTL? I always assumed that it would be LTR. However, divide reverses numerator and denominator so how do we handle a complex chain of operations like the above example?

What about the following operations?

$$5\div5\div5\div5\div5\div5\times5\div5\div5\div5\div5\div5 \tag2$$

and $$5\div5\div5\div5-5\div5\div5\div5\div5\times5\div5\div5+5\div5\div5 \tag3$$

Here is the solution in the book:

enter image description here

Real Noob
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    Thanks @Blue. I couldn't figure out how to write the expressions more elegantly. They look much better now. :) – Real Noob Jul 03 '21 at 21:19
  • Could anyone reading this question please explain why the question is getting downvotes? it has received two so far. :) – Real Noob Jul 03 '21 at 21:23
  • It is common practice to do addition and subtraction left-to-right, multiplication and division left-to-right, and exponentiation right-to-left – Henry Jul 03 '21 at 21:29
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    I try to avoid such questions by use of parentheses. – herb steinberg Jul 03 '21 at 21:32
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    Not only is the book's answer unusual, the very fact that they ask such a question is a strong indicator that the best use of this book is to put it in the bin for paper recycling. Then at least something useful might be made from it. – David K Jul 03 '21 at 21:35
  • So, is the solution posted in the textbook wrong? – Real Noob Jul 03 '21 at 21:35
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    The convention is that $\times$ and $/$ are done first from L to R, followed by $+$ and $-$ from L to R. And exponentition is done from L to R before $\times$ and $/$. And a tower of exponents is done from the top, downwards. So $1+3^{2^3}/81=1+((3^8)/81)=82.$ – DanielWainfleet Jul 03 '21 at 21:37
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    @DavidK: It's sad that students are being taught that the order of operations are eternal truths of mathematics, rather than helpful conventions. – Joe Jul 03 '21 at 21:46

2 Answers2

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As far as I know, when dealing with consecutive multiplication and division, one can simply go from left to right. This means that your example problem is solved as follows: $$ \begin{array} \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5}\div 5 \div 5 \div 5 &= 1 \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5}\div 5 \div 5 \div 5 \\ &= 5 \div \frac{1}{5} \div \frac{1}{5}\div 5 \div 5 \div 5 \\ &= 25 \div \frac{1}{5}\div 5 \div 5 \div 5 \\ &= 125\div 5 \div 5 \div 5 \\ &= 25 \div 5 \div 5 \\ &= 5 \div 5 \\ &= 1 \\ \end{array} $$

If you wanted to, you could turn the division into multiplication as you suggested:

$$ \begin{array} \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5}\div 5 \div 5 \div 5 &= \frac{1}{5} \times 5 \times 5 \times 5 \times 5 \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \\ &= \frac{5 \times 5 \times 5 \times 5}{5 \times 5 \times 5 \times 5} \\ &= 1 \end{array} $$

Kman3
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  • Thanks @Kman3. :) The answer in the book to this question was 1/25 and they went RTL to solve this question. So, are they wrong? – Real Noob Jul 03 '21 at 21:28
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    @RealNoob I don't think you can go from right to left when you're dividing. For example, $2 \div 3$ and $3 \div 2$ are not the same. I'm not sure where your textbook gets $\frac{1}{25}$ from; when I go from right to left as you suggested, I get $625$. – Kman3 Jul 03 '21 at 21:30
  • I have added a screenshot of the solution here. :) – Real Noob Jul 03 '21 at 21:32
  • @RealNoob Oh. That was sloppy on the part of your textbook: it isn't following the order of operations, instead picking and choosing where to start without proper parentheses added. – Kman3 Jul 03 '21 at 21:33
  • They seem to be going all the way RTL from what I can tell? :) – Real Noob Jul 03 '21 at 21:34
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    @RealNoob Ah, I misinterpreted what you meant by "right to left". Yes; it appears your textbook is doing $a \div (b \div ( c \div d ))$ when they've written $a \div b \div c \div d$. They still haven't solved the problem through the standard order of operations, which would be $((a \div b) \div c)\div d$, the other way (i.e. from left to right). – Kman3 Jul 03 '21 at 21:37
  • Some kids are more likely to trust a book than what I say if they were to ever come across such a question. To them, I would be a stupid instructor who couldn't solve a simple division question. Typos in a book are understandable but entire questions solved incorrectly in a highly reputed book will shift the blame on me. – Real Noob Jul 03 '21 at 21:43
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    @RealNoob Definitely. However, you are not to blame; I have never liked these kinds of questions. To me, they only serve to confuse students rather than teach them material that will be useful to them later in higher levels of mathematics. If you are an instructor and you have the ability to do so, I would focus on different material that is less ambiguous so as not to confuse your students. – Kman3 Jul 03 '21 at 21:46
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    @RealNoob If, however, you are obliged to follow a curriculum as prescribed by your jurisdiction, you can just tell the students that the order is (1) parentheses (2) exponents (3) multiplication and division, going from left to right (4) addition and subtraction, going from left to right. There are other rules about exponents that have been mentioned in the comments, but I think you can stick to those guidelines without any problems. The internet can also be useful if you want to check the consensus on order of operations. – Kman3 Jul 03 '21 at 21:48
  • Thanks for the solution @Kman3. Some day one of these students could bring a question like this to me while they were practicing from some other material. So, there is always a possibility of me looking stupid for no reason. – Real Noob Jul 03 '21 at 21:49
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    @RealNoob I'm not an instructor, though I understand your concerns. The best that you can do when things like this come up, in my opinion, is to tell the students how you got your differing answer rather than simply stating that the textbook is wrong and moving on. That way, they will be more convinced by your reasoning. And if they wish to demonize you for making mistakes (and everyone makes mistakes), then they too should be prepared to be demonized by their instructor when they make mistakes. – Kman3 Jul 03 '21 at 21:52
  • Hehe thanks @Kman3 that makes sense. :) – Real Noob Jul 03 '21 at 21:53
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    @RealNoob You're welcome. While I'm at it, I will refer you to a Stack Exchange site for math educators like yourself if you want to ask questions in the future specifically about mathematics pedagogy and its challenges. – Kman3 Jul 03 '21 at 21:55
  • Thanks @Kman3 that website looks helpful. :) – Real Noob Jul 03 '21 at 21:57
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$\begin{align} \frac15\div\frac15\div\frac15\div\frac15\div\frac15\div5\div5\div5 &= \frac15 \times 5 \times 5 \times 5 \times 5 \times \frac 15 \times \frac 15 \times \frac 15 \\ &\phantom{=} \text{(Four 5's above and four 5's below)}\\ &= 1 \end{align}$

Steven Alexis Gregory
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