## An Algebra Text Book

I am usually disappointed with American math text books. I have had an underwhelming experience with them. Often I open a book and in the next 15 minutes, I find a mistake, a typo, a misguided explanation, sloppiness, a misconception or some other annoyance.

I was pleasantly surprised when I opened the book Introduction to Algebra by Richard Rusczyk. I didn’t find any flaws in it — not in the first 15 minutes, and not even in the first hour. In fact, having used the book many times I have never found any mistakes. Not even a typo. This was disturbing. Is Richard Rusczyk human? It was such an unusual experience with an American math book, that I decided to deliberately look for a typo or a mistake. After half a year of light usage, I finally found something.

Look at problem 7.3.1.

Five chickens can lay 10 eggs in 20 days. How long does it take 18 chickens to lay 100 eggs?

There is nothing wrong with this problem. But the book is accompanied by the Introduction to Algebra Solutions Manual in which I found the following solution, that I’ve shortened for you:

The number of eggs is jointly proportional to the number of chickens and the amount of time. One chicken lays one egg in 10 days. Hence, 18 chickens will lay 100 eggs in 1000/18 days, which is slightly more than 55 and a half days.

What is wrong with this solution? Richard Rusczyk is human after all.

I like this book for its amazing accuracy and clean explanations. There are also a lot of diverse problems in terms of difficulty and ideas. Richard Rusczyk has good taste. Many of the problems are from different competitions and require inventiveness. I like that there are a lot of challenge problems that go beyond the boring parts of algebra. Also, I like that important points of algebra are chosen wisely and are emphasized.

This book might not be for everyone. It doesn’t have pretty pictures. It doesn’t have color at all. This is not a flaw for a math book. The book concentrates on ideas and problems, not entertainment. So if you’re looking for math entertainment, you’ll find it on my blog. For solid study, try Richard Rusczyk’s books.

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1. #### Andrew MW:

I think it’s the nature of egg laying…

I solved it by thinking…
(a) a chicken can lay an egg in 10 days.
(b) therefore, in 50 days, 18 chickens can lay 90 eggs.
(c) the last 10 will take another 10 days regardless – i.e. 60 days all up.

The 18 chickens can’t suddenly “rush out” the last 10 eggs in 5.5 days!

2. #### Sue VanHattum:

Ahh, but if they’re laying on different days, would it work out?

Actually, the biggest problem I see is that those chickens would be stew. Normal chickens lay an egg a day, or in the winter somewhat less. But one egg in 10 days? She’s not doing her share…

3. #### Xamuel:

Re: Pictures and Color

I’ve always said: no matter how many irrelevant pictures of race cars and motorcycles you stick in your math book, people are still going to assume it’s a tome of black magic. You might as well make it *look* like a tome of black magic, because at least that way it’s *cool* 🙂

Of course, this begs the question: is that a soccer ball on the cover? ugh..

4. #### Andrei Zelevinsky:

Seems to be a nice book. I am thinking to give it as a B-Day gift to a bright third-grader. Btw, you might want to replace the link to Amazon.com with the one to their own bookstore: https://www.artofproblemsolving.com/Store/viewitem.php?item=intro:algebra.

5. #### misha:

Maybe the author of the book meant an average chicken by a chicken. Each of 18 chickens probably will have her own schedule, but in 5 days they lay 9 eggs, so the solution given in the book looks fine to me. Again, it’s a matter of interpretation. Actually, Sue VanHattum has already pointed it out.

To Xamuel: is that a soccer ball on the cover? ugh.. Looks like one, it must be the logo of the series, I guess it is supposed to represent one of the Platonic solids.

6. #### colorblind:

Sue is surely right. Ever seen Chicken Run? Pot pies for everybody!

More to the question, I think Andrew touches on the real problem, but mis-states it. The author assumes the chickens are continuously laying eggs as in if a chicken can lay an egg in 10 days then in 5 days it will lay half an agg. That’s a sore, sore chicken.

7. #### Geneal:

There’s another wrinkle that I think I remember from Geoffrey Mott-Smith’s book of puzzles. He stipulates that 10 sheep can jump over a fence in 10 minutes and asks how many sheep can jump over a fence in an hour. The answer, of course, is not 60, but 55. From the first jump to the tenth jump takes 10 minutes, so the interval between jumps is 10/9 minutes, and one hour can be divided into 54 such intervals. But there’s a jump at the beginning as well as the end, making 55 in all.

Applying this line of reasoning to the chicken and egg problem, and assuming that the chickens lay eggs in rotation, the interval between successive egg layings by any one chicken is 100/9 days. Now we have 18 chickens laying in rotation, so the interval between egg layings by *any* chicken is 100/162 days. If we start measuring as the first egg is laid, 100 eggs will have been laid at the end of 99 such intervals, or 9900/162 = 61.1111… days, or 61 days 2 hours 40 minutes.