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sure accuracy and rapidity in combining numbers, and promptness in solving mentally such problems as are likely to occur in the every-day business affairs of life.

The plan of this work is intended to aid in the accomplishment of these important results. It unfolds in a natural order the principles underlying every topic presented, and leads by easy gradations and logical analyses to deductions more complex and abstract. The applications of principles are illustrated by numerous and varied examples arranged progressively; the problems and exercises are carefully and systematically classified; reviews combining new principles and methods of solution with those previously learned are placed at the close of almost every section and chapter.

Models of analysis are given, brief and logical in form and correct in expression. While they are offered as the result of long experience in the school-room, it is not expected that they shall be rigidly adhered to; but they will aid pupils to give additional analyses in their own words. The greatest care should always be taken to secure the strictest accuracy in the statement of the various steps by which fundamental truths are evolved; equal care should be taken to create and stimulate in pupils habits of self-dependence; to train them to rely upon principles and reason, rather than upon mere forms; and to think, compare, and investigate for themselves; for knowledge of this subject becomes useful and available only when they can perceive the conditions of any given problem, understand the relations existing between those conditions, and know how to apply principles in solutions necessary to obtain required results.

Answers to the more difficult problems, together with additional models of analysis and forms of solution are given in the Key; in it are offered also some suggestions as to methods of teaching this important branch of Arithmetic, which, it is hoped, will be found useful to the teacher.

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I. PROPERTIES OF NUMBERS.

FACTORS, OR DIVISORS
COMMON FACTORS, OR DIVISORS
COMMON MULTIPLES, OR DIVIDENDS.

59 59 62 65

CHAPTER III.- Fractions.

I. COMMON FRACTIONS
II. REDUCTION
III. ADDITION
IV. SUBTRACTION .

V. MULTIPLICATION
VI. DIVISION.
VII. RELATION OF NUMBERS
VIII. DECIMALS

NOTATION AND NUMERATION
REDUCTION.
APPLICATIONS

67 75 84 87 91 99 107 115 115 122 126 GENERAL CASES.

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II. APPLICATIONS

PROFIT AND Loss

COMMISSION

INTEREST

DISCOUNT

INSURANCE.

STOCKS

MISCELLANEOUS PROBLEMS .

158

. 158

161

164

168

. 171

173

. 176

THE

MODEL

MENTAL ARITHMETIC.

CHAPTER I.

INTEGERS.

Section I.

Definitions. 1. A Unit is one, or a single thing. Thus, one boy, one book, or one is a unit. 2. A Number is a unit, or a collection of units. Thus, one chair, ten years, two, or twelve is a number.

3. The Unit of a Number is one of the collection forming that number.

Thus, one man is the unit of six men; one is the unit of six.

4. An Integer is a number whose units are whole or undivided.

Thus, one, three, seven days, twenty miles are integers.
Integers are also called Whole Numbers, or Integral Numbers.

5. A Concrete Number is a number whose units are named. Thus, one yard, nine pens, twelve eggs are concrete numbers.

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6. An Abstract Number is a number whose units are not named. Thus, one, nine, twelve, twenty are abstract numbers.

ny. Similar Numbers are numbers whose units are alike.

Thus, five feet and eight feet are similar numbers; also, three ones and six ones; four tens and five tens; six and nine.

8. Dissimilar Numbers are numbers whose units are not alike.

Thus, five feet and eight pints are dissimilar numbers; also, three ones and six tens; four tens and eight hundreds.

9. Arithmetic is the science of numbers and the art of computing by them.

Note. - As a science, arithmetic treats of the principles, properties, and relations of numbers. As an art, it teaches how to apply the principles of numbers to practical purposes, or to business affairs.

10. Oral or Mental Arithmetic is the process of solving problems without the aid of written characters.

11. Written Arithmetic is the process of solving problems with the aid of written characters or figures.

12. A Problem is a question which requires a solution.

13. A Principle is a general truth upon which a process of solution is based.

14. An Analysis is a statement of the different steps in the solution of a problem.

Exercises.

1. How many units in one? In three? In nine? In one day? In two feet? In six pounds ?

2. Two is a collection of how many units ? Four? Six months ? Five dollars ? Seven days? Nine quarts ?

3. What is the unit of two cents? Of four miles? Of three? Of five? Of eight inches? Of twelve? Of nine tons ?

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