Interest, Single and Compound, Fellowship, Single and Double, Barter, Exchange, Allegation, Position, &c. are but so many applications of these principles, as I shall show when I come to treat of Proportion. By a clear understanding of the principles of these numbers, the higher branches of the Mathematics are also rendered easily accessible. Algebra and Fractions are the same in their manner of operation; the only difference is, that in the one rule we use figures to represent quantities, and in the other we use symbols. 275. I hope I have succeeded in making the student see something of the importance and utility of this branch of arithmetic, and to induce him to commence its study with attention, perseverance*, and industry. He must summon to his assistance all his powers of thought; his mind, as well as his fingers, must be active. The whole man must be engaged in the object. Let me beseech him by no means to slur the matter over, for a little attention, as he proceeds, will save a world of trouble afterwards. 276. In former parts of this work I have shown the advantage of performing operations by sensible objects, at the same time they are performed abstractedly by figures. Now if this mode of performing operations is an advantage in whole and compound numbers, it is more so in fractional ones. These numbers being naturally more intricate, they require additional means of rendering them simple and intelligible. When a boy at school myself, and engaged in the working of fractions, I derived more information, as to the nature and manner of operation of these numbers, from a common measuring two foot rule, than from any other source; and having found it so useful myself, I recommend it to others. 277. This instrument is so generally known that any explanation of it may, perhaps, be thought superfluous; but in order to show its application in explaining and working Fractions, some description of its parts will be requisite. This instrument, as every one knows, is formed of two pieces of thin boxwood, each piece a foot long, and united by a hinge at one end for the convenience of folding. Each piece is divided into twelve divisions, called inches, consequently the whole rule contains twenty-four inches. Each inch is further divided into eighths; these eighths may be considered as eighths of an inch, or forty-eighths of six inches, or ninety-sixths of a foot, &c. Observe the following Table of Equivalents. * Perseverance, firmness, resolution, steadiness in the pursuit of an object; from Persevero, to persist, to hold out. 1 2 ลง 34 5 6 7 819 10 | 11 | 12 Inch Inch Inch Inch Inch Inch Inch Inch Inch Inch Inch Inch 271. By having a fraction from to of 1 inch given, by the above table we have its equivalent fraction of any number of inches from one to 12. The table is constructed on the same plan as the Multiplication Table. The first column and the top line are the indices: thus, if we would find what part of 5 inches of 1 inch is, we look in the first column for g, and on the same line, under 5 inches, we find o, which shows that f of 1 = % of 5. 40 279. Let the student carefully consider the above tables and test their correctness by the application of the scale alluded to above. 280. Let the student take his scale, and answer the following questions from it. What part of 8 is the 2 of 3* ? of 2 ? * Half of 3 inches is 1% inch, or 3 half inches. One half-inch is the sixteenth part of 8 inches; therefore three half-inches are three sixteenth parts of 8 inches, that is, of 3% of 8. L 281. of of Consider the of 9 inches is 6 of 1, is what part of 3? one foot; then of a foot is 9 inches, inches, of 6 inches is 1 inch, and 1 inch is of 1 foot must be of 3 feet; therefore ofofof 1 = of 3. of a foot, and 282. For the Abstract Rule of Reducing Compound Fractions, such as the above, see a few paragraphs forward. 1. Are fractions generally represented as presenting great difficulties? 2. Why do fractions present such difficulties to beginners? 3. From what do the perplexity and obscurity in which fractions are involved arise? 4. Are persons, who have learned arithmetic merely by rule, without reference to principles, apt to suppose that, on entering on a fresh rule, they are entering on a different subject? 5. Are the principles of the different rules in arithmetic essentially different? 6. Is there any thing peculiarly difficult in fractions ? 7. Should difficulties present themselves, what course should the student pursue to overcome them ? 8. What is required on the part of the student in order to enable him to become an expert arithmetician? 9. Has the study of fractions a tendency to strengthen and invigorate the mind? 10. What are fractions, and how do they originate? 11. What determines the value of a fraction? 12. What does the numerator of a fraction describe? What does the denominator describe ? 13. In what cases do fractions exceed unity, and what are such fractions then called? 14. When are fractions less than unity, and what are such fractions then called? 15. What is a mixed number? 16. When a fraction exceeds unity, what process will give its value? 17. How are whole numbers represented in fractional terms? 18. When one number is intended to be divided by another, how should such numbers be represented in fractional terms ? How should the quotient resulting from such division be written down? 19. May all fractions be considered as representing a divisor and a dividend? 20. Do fractions possess the principle of shortening operations? 21. Is much of the science of arithmetic exhibited in fractions? and are the principles of this department of arithmetical science much the same as those of Proportion, Interest, Exchange, and Algebra ? 22. Let the student show how a two-foot rule may be applied in explaining fractions; let him also be examined on the fractional tables of equivalents. CHAPTER IX. REDUCTION OF FRACTIONS: 284. By REDUCTION of FRACTIONS is meant the changing of them from one denomination to another. This change of form is frequently necessary for the purposes of calculation. First, there is the reducing them from large and inconvenient terms to their most simple and convenient form, as the reducing 18945 to the simple form of . Secondly, the reduction of whole or mixed numbers to improper fractions, and of improper fractions to whole or mixed numbers. Thirdly, the reduction of compound and complex fractions to simple ones. Fourthly, the reduction of fractions of one integer to their equivalent fractions of another integer. And, fifthly, the reduction of fractions of different denominations to one common denomination. All this is very easy work, but it requires a little thought and a little care to preserve the proper proportion between the numerator and denominator. Of the importance of this essential, the student must by this have become fully impressed. CASE 1. To Reduce Fractions to their Least Terms. 285. It has been shown at paragraph 265, that the value of a fraction depends on the proportion which one of its terms bears to the other; therefore the value of a fraction is not altered, if both its terms be multiplied by the same numbers |