Ray’s Arithmetic

I have seen many people struggle with arithmetic while I was in school, college, and also when I was working. Most people—both children and adults—have not developed the capability to do calculations in their head; they need a calculator or pen and paper to do even simple calculations.

When I was in studying in school, I used to watch the sales personnel at grocery shops and markets, and waiters at the not so fancy restaurants performing arithmetic calculations with ease and speed. I was amazed by their memory and the speed at which they could calculate. They knew the rate of the items, the quantity of each item ordered (in a list of 25 or more items) and would call out loud so that the shop owner could write the bill and collect the money. “Sugar 2 Kg, 10 rupees; Wheat flour 1Kg, 6 rupees; mustard 250gm, 1.50 rupees; Colgate paste large 1, 6 rupees; Pears Soap 3, 12 rupees…” and so on they will call out and finally give a summary: “19 items, total 84.50 rupees.” The shop owner would check it with the calculator and every time the salesman would be right. I often wondered how a person, who must have studied may be up to the fourth standard, could do calculations with such speed and accuracy.

While I was studying in the fifth standard we had a good maths teacher. He was very good at arithmetic. It was he who introduced us to speed arithmetic. Speed arithmetic is easy. You don’t have to be a rocket scientist to master it. Only thing you need is to learn a few rules and then practice, practice and practice more.

One requisite for learning speed arithmetic and perform arithmetic operations is the number sense. You should learn things like 0.25 is larger than 0.20, a natural number when multiplied by another will give a product that is greater than both the numbers, a natural number when multiplied by a fraction or decimal will give a number which is smaller than the natural number, a number divided by another number will give a number less than the number that is being divided whereas if it is divided by a proper fraction or decimal, the result will be larger number and so on. The number sense is something that you gain by learning the rules and develop as you practice. As you practice more, you will develop your own shortcuts and tricks to compute faster.

The number sense is very valuable as it will help in calculating faster and to realize when you have made a mistake. Today, the students lack this number sense and the main reason is lack of practice. Another reason is that the teaching method has changed so much that even for performing a simple addition, subtraction, multiplication, or division the calculation method is too complicated. For addition and subtraction the ‘carrying’ and ‘borrowing’ are made so complex that it increases the chances of making mistakes.

When I was working in Pond’s (India) Ltd. at Pondicherry, one of my hobbies was visiting the bookstores during my spare time. There were some very good used bookstores. On one such visit, I saw a set of 6 books titled ‘An Introduction to Ray’s Eclectic Arithmetic.’ These were very old books and the shop owner told me that he had it for many years and I could have it for 15 rupees. I was intrigued by the title as I have never heard of Eclectic Arithmetic and wanted to know what it was. That was my first encounter with Ray’s Arithmetic books. I found the books interesting and the approach novel. But the books were so old that after a little handling the pages began to crumble. So, I dumped those books.

I teach my nephew and one day while I was teaching, I remembered the Ray’s Arithmetic books that I had discarded. So, without much hope, I searched the Internet and to my astonishment, the books were available. When I first encountered the books back in 1991, I didn’t have any idea who Joseph Ray was or how popular his books were. Joseph Ray was a mathematician, doctor, author, and educator. He wrote a series of six books that were published in 1834 as “An Introduction to Ray’s Eclectic Arithmetic.” The series soon became the base of arithmetic and algebra textbooks for American mathematics across the nation. By 1913, the series sold an estimated 120 million copies and annual sales were reported to exceed more than 250,000. Then for some reason their popularity waned and they become out-of-print.

The books that were out-of-print were reprinted by Mott Media in 1985 and are available as a set of eight books—Ray’s New Arithmetic. The eight books are:

  1. Primary Arithmetic
  2. Intellectual Arithmetic
  3. Practical Arithmetic
  4. Higher Arithmetic
  5. Test Examples in Arithmetic
  6. Key: Primary, Intellectual, Practical
  7. Key to Higher Arithmetic
  8. Parent-Teacher Guide

The books are organized in an orderly manner around the discipline of Arithmetic itself. They present principles and follow up each one with examples, which include difficult problems to challenge even the best and brightest. Students who do not master a concept the first time can return to it later, work the more difficult problems, and master the concept. Thus these books provide a complete arithmetic course to study in school and to prepare for competitive examinations.

One main advantage of this book is that it teaches the students addition, subtraction, multiplication, and division up to 12 using tables. In school, I have only studied multiplication tables. Once the student has learned the 48 tables (12 each for the 4 arithmetic operations), his calculation speed and accuracy increases dramatically. What is learned in each table is reinforced using the countless exercises.

Another useful feature is that the problems are specified as complete sentences. One of the main difficulties that I have found in kids is that while they are able to answer the questions that are given as numbers and signs (like 4 + 3, 50 – 30, 12 x 4, etc.), they find it difficult to answer the number stories. Number stories are questions where the questions are asked as they occur in real life. For example, “Ram had Rs. 30,000 with him. He gave Rs. 6000 to his sister. He bought 4 cows each worth Rs. 3000. He also bought 5 shirts each for Rs. 400. If he gave one fourth of the remaining amount to his friend, how much money is left with him?” Students often find it difficult to determine the arithmetic operation they have to perform when faced with a number story. In Ray’s books, almost all the problems are given as number stories starting with very easy ones and moving on to difficult ones. This helps the student to tackle number stories with more confidence.

The new edition has retained the examples and problems as such. This helps in retaining the charm of a former era and gives an idea about the prices of things more than 150 years ago. Given below are some examples from the second volume—Intellectual Arithmetic.

  1. If 12 peaches are worth 84 apples, and 8 apples are 24 plums how many plums shall I give for 5 peaches?
  2. A gentleman meeting some beggars found that if he gave each of them 3 cents he would have 12 cents left. But if he gave each of them 5 cents he would not have money enough by 8 cents. How many beggars were there?
  3. A hare is 10 leaps before a hound, and takes four leaps while the hound takes 3. But two of the hound’s leaps equal three of the hare’s. How many leaps must the hound take to catch the hare?

I hope you would enjoy teaching the kids with these books and your kids would enjoy learning arithmetic using Ray’s method. I wish you all the very best. For a little inspiration to master arithmetic watch Arthur ‘Mathemagician’ Benjamin’s amazing performance at TED.


  1. Paresh said,

    August 31, 2012 at 12:30 am

    This is quite a blast! Something like Vedic Maths… I am not very good at maths beyond the basic division. Will try this out. Thanks 🙂

  2. Alexis Leon said,

    August 31, 2012 at 8:49 pm

    Paresh: Thanks Paresh. Some speed arithmetic systems are The Trachtenberg Speed System of Basic Mathematics, Arthur Benjamin’s Secrets of Mental Math, Abacus System, Chisanbop and also Vedic Mathematics.

  3. Maya said,

    August 31, 2012 at 11:10 pm

    Nice and informative post. I face the same problem–my son is finding it difficult to solve the number stories. I will definitely try the Ray’s method to teach him.

  4. Bindu said,

    September 1, 2012 at 12:01 am

    This post couldn’t have come at a better time. Georgie’s school doesn’t have text books and the absent minded mother that I am, have no clue what he is taught, what he knows and what he doesn’t. I have been on the look out for something like this, where it is simple, at the same time challenging as well. Hope these are available on flipkart.
    Thank you so much!

  5. Tme said,

    September 1, 2012 at 4:52 am

    That was a wonderful post,Alexis.

  6. Alexis Leon said,

    September 1, 2012 at 11:13 am

    Maya: Thanks Maya. I wish you all the best with teaching. I am sure it will work.

    Bindu: Bindu, these books are ideal for you. The books explain the principles with solved examples. Then there are lots of exercises, starting from easy ones and moving to more challenging ones. The kids would have no difficulty in learning using this book. The answer key is another useful feature. I wish Georgie all the best and if mom can set up a reward system for completing the exercises, I am sure he will be more than willing to take the challenge :-). Yes the books are available on Flipkart (ISBN: 9780880620505).

    TME: Thanks Thulasy.

  7. Sheeba said,

    September 1, 2012 at 2:10 pm

    Excellent post ! Thanks a Lot. Congrats too.
    Very useful for the people like me,
    who are addicted to calculator 🙂
    Hope u r fine..

  8. Alexis Leon said,

    September 2, 2012 at 8:31 pm

    Sheeba: Thank you Sheeba. The availability of calculators have made us lazy. Throw the calculator away and you will regain your mental capabilities :-).