Joseph Ray believed that the ultimate objective in teaching was to develop high moral character. It is said that he would often interrupt a lesson to call attention to some fault of a student and then tell a story about the need for persons with high morals. Story problems in his textbooks portray honest, hard-working men and women on the frontier, plowing fields, planting and harvesting crops, building walls, or buying and selling goods. Problems about boys and girls characterize them as industrious and generous, sharing food and possessions with each other. In one problem a boy receives a reward for returning a purse to its rightful owner.

The higher books all have Keys which are included. The Keys are all keyed to (excuse the pun) the "Article Numbers" in the main textbooks. "Articles" are the same as "Lessons" to us. The Keys contain answers to the more difficult problems in the books. It is suggested that students use these to check their work. Students should move from section to section as they master the problems in that section. Generally, a section takes a day to do, however, some only take a few minutes and some have many problems to work. They are not of uniform difficulty. 

Because these books are delivered on CD and printed by you they are normally printed much larger than the originals, making them that much easier to use and friendlier to younger eyes.  If you would like to have a copy of the smaller printed versions you can get them from Kings Harvest.  They also have sample problems and pages as well as a nice description of each book.

Ray's Arithmetic - Basic Math contains the following books: 
 

Suggested Grade Level	Books	Description
Grade 1  Grade 2	Primary Arithmetic	Simple mental lessons and tables. For little learners.
Grade 3 Grade 4	Intellectual Arithmetic Key to Intellectual Arithmetic	The most interesting and valuable Arithmetic extant.
Grade 5 Grade 6	Practical Arithmetic Key to Practical Arithmetic Test Problems for Practical Arithmetic Dubb's Arithmetic Problems (for Ray's Arithmetic)	A full and practical treatise on the inductive and analytical methods of instruction.
Grade 7 Grade 8 (Grade 9?)	Higher Arithmetic Key to Higher Arithmetic	A simple, thorough, and progressive elementary treatise.

The Ray's books never mention grade level. Advanced students will finish Higher Arithmetic at the end of the 7th grade. Others will take longer.

Intellectual Arithmetic. The student probably needs some help here as well, but should be allowed to reference the Key as the Key shows how problems are worked as well as providing the answers. Working through this book is relatively straightforward the student learns to add, subtract, multiply, divide, about fractions, interest and percentage. A problem at the end of the book:
James sold 10 oranges for 40 cents and gained 33 1/3 %. How much did each orange cost? Suggested for grades 3 & 4, but your mileage might vary. It is 140 pages.

The Practical Arithmetic book has two problem books to go with it as well as a Key. The problem books are Test Problems for Practical Arithmetic and the other is Dubb's Arithmetic Problems for Ray's Practical Arithmetic. The Dubbs book was published a decade after the others, but it is an actual part of the series and good workbook (with answers in the back) and it is suggested that the student work all the problems in both books. This amounts to some 8000 problems over 2 years, which, assuming 200 school days in each, is 20 problems a day - a reasonable number. Certainly more than their peers are doing! Both of the problem books are keyed to the "Article Numbers" in the main textbook, so it is easy to keep all the books synched up. However, understand the student will work out of 4 different books. Or at least 3, assuming he doesn't do Dubbs.

Practical Airtmetic is 336 pages. Reviews adding, subtracting, etc., but raise the bar quite a bit: Compound numbers, discount rates, exchange rates, taxes, ratio, proportion, square root, and progression. What I absolutely adore about this book is that it is all real-world story problems. For instance, in exchange rates, it tells you various countries, their currency and the exchange rate, and then asks a bunch of questions. The entire book has that flavor to it – it is totally intertwined into the real world. Certainly, some of it is archaic and your child might ask you what you what a "spinning wheel" is. Of course, don't be surprised if they also ask you what it means to "wind a watch."

Higher Arithmetic is a challenging book which covers a lot of material. There was no Test Problems book ever produced for it, my theory being that the problems in the book more than suffice. It was originally intended for "high school and into college." However, that was when most schools only had 10 grades! It is 408 pages. In many ways, this book is more than a math textbook; it is handbook for living life and how you use math in the ordinary course of your daily life. For many students, this is as high as they would have gone in regular school. This book contains normal math lessons in a logical progression but applied to real-world situations. For instance, in the percentages section, you study Profit & Loss, Stocks & Bonds, Insurance, and so on. You learn about writing checks, promissory notes, figuring how much water a boat displaces, double-entry bookkeeping, and dozens of other useful math-related facts. But it has all the SAT stuff as well. Cube roots, greatest common denominator, etc.

FEW works on Intellectual Arithmetic have received more unqualified approbation, and a more extensive patronage, than this, which, for several years, was published under the title—Bay's Arithmetic, Part Second.

The numerous editions demanded have again rendered necessary a renewal of the plates; this has afforded an opportunity for remodeling the work.

Many important improvements have been introduced, with a design to impart completeness, and give a concise and progressive course of arithmetical analysis.

The volume in its present form, embraces:

1st. Exercises on the primary principles, and their applications; together with models of analysis in the shape of solutions.

2d. A progressive and comprehensive presentation of Fractions, intended to render the subject intelligible and attractive.

3d. General Reviews, designed to test the pupil's knowledge of principles, preparatory to the applications of mental analysis which follow.

4th. Reduction and Percentage, with the application of the latter to Profit and Loss, Commission, Insurance, Interest, and Discount.

The value of Intellectual Arithmetic is so highly appreciated, that little need be said in its commendation; when properly taught, it is one of the most useful and interesting studies in which pupils can engage. By its study, the pupil is taught to reason, to analyze, to think for himself; while it imparts confidence in his reasoning powers, and strengthens the mental faculties.

CINCINNATI, May, 1877.

THE solution of an example in Mental Arithmetic is herein regarded as a direct and natural response to the requirements of the problem; it aims to be a plain and simple statement of the successive steps leading to the final result or answer. The data afforded by the example are repeated in the solution no further than is necessary for its clear statement.

Every different class of examples is furnished with a solution; in a few instances, more than one solution is given, the succeeding solutions offering a briefer method. No solution is repeated; and, when a solution, in whole or in part, occurs in a subsequent solution as a portion of the argument, its result is given as briefly as possible.

Of course, there is nothing to prevent the teacher from elaborating any solution, or any portion of it, to the fullest extent he sees fit; and it may often be desirable so to do. Furthermore, those who wish can, with facility, apply the "Four Step Method" for the three step, of which each solution will furnish a clear statement.

It is obvious, that in a text-book the publishers should not occupy space with any thing more than the essentials of a solution.

CHANGES in the methods of instruction in our schools and in the modes of transacting business have made it necessary to revise Ray's Practical Arithmetic.

No other work on Arithmetic ever had so extensive use or wide spread popularity. Teachers every-where, throughout the length and breadth of the land, are familiar with its pages, and millions of pupils have gained their arithmetical knowledge from the study of its principles. More than ten thousand editions of it have gone forth from the press.

In view of these facts, it has been the constant aim in making this revision to preserve carefully those distinctive features of the former editions, which constituted the peculiar philosophical method of its learned author, viz.:

1st. Every principle is clearly explained by an analysis or solution of simple examples, from which a Rule is derived. This is followed by graduated exercises designed to render the pupil familiar with its application.

2d. The arrangement is strictly philosophical; no principle is anticipated; the pupil is never required to perform any operation until the principle On which it is founded has first been explained.

The changes made fall naturally under two heads: (1) those that adapt the book better to the advanced methods of instruction; (2) those that exhibit present methods of computation in business.

In the first place, special attention is invited to the beauty and elegance of the typography. The different matter of the volume the definition, the solution, or the rule, is at once clearly indicated by a difference of type. A running series of articles, with numbered paragraphs, enhances the convenience of the text-book for recitation and for reference.

The analytic solutions and written operations have been carefully separated. All obsolete Tables of Weights and Measures, such as Beer Measure and Cloth Measure, and all obsolete denominations, such as drams, roods, etc., are discarded. The Metric System of Weights and Measures is presented in accordance with its now widely extended usage, and is assigned its proper place immediately after Decimals.

A few subjects, such as Factoring and the principles of Fractions, have been entirely rewritten, and in many instances the definitions and rules have been simplified. The subject of Percentage has been much expanded, and an endeavor has been made to systematize its numerous applications; many novel and interesting features, both of subject-matter and classification, will here be met with for the first time. The subjects of Interest and Discount have received that careful attention which their importance demands.

The publishers desire to express their thanks to the many teachers whose suggestions and corrections are embodied in the present edition. Especial mention is due Prof. M. W. Smith and Mr. A. P. Morgan for many valuable features of this revision.

In conclusion, the publishers wish to reiterate that the object throughout has been to combine practical utility with scientific accuracy; to present a work embracing the best methods, with all real improvements. How far this object has been secured is again submitted to those engaged in the laborious and responsible work of education.

CINCINNATI, August, 1877.