that of these were just of what he had in both. How many sheep had he in the second pasture? Prove. 5. Two boys, talking of their ages, one said he was. 9 years old. Well, said the other, of your age is exactly of mine. What was his age? Prove. 6. A farmer and his son went out one day to look after his sheep, which were in two separate fields. They counted 35 in one pasture, and the farmer told his boy that of these were just of what were in his other pasture. How many had he in both pastures? Explain the process, or prove. 9 10 7. A farmer had sheep in three different pastures. In the first he had 100. In the second he had of of what he had in the first. [How many are of 100? Then how many In the third he had of 1⁄2 of what he had in the second. How many sheep had he in all? Explain the process. are of that?] 8. A gentleman had three sons. Being asked the age of the youngest, he replied, that the age of the eldest was 16; the age of the second son was of of that of the eldest; and that the age of the youngest was of that of the second son. What was the difference in age between the eldest and youngest son? Explain the process. 9. A young lady, who was 16 years of age, having asked her female friend how old she was, received the following reply: If of your age were added to your age, the sum would exactly show how old I am. How much older was she than the lady who put the question? Explain the process. CHAPTER III. FRACTIONS OF UNITY. SECTION I. First Principles. [IN the following exercises, let the teacher write the questions on the black-board, and add the answers as fast as they are announced by the class, in the following form, viz. : 2 X 16 1. Which number is the numerator of a fraction? Which the denominator? What are both called? How much is 2 times? How many times does contain? Ans. Two times. How much is 3 times? How many times does 3 contain? How much is 5 times? How many times does contain? [Point to the black-board.] Will multi1/6 plying the numerator of a fraction by any number, then, always make the fraction so many times greater? Remember, then, that multiplying the numerator multiplies the fraction. 16 2 16 2. Divide by 2; in other words, what is the half of 12? How many times is contained in 1? What is the third part of ? in other words, divide by 3. How many times is contained in 12? [Point to the black-board.] Will dividing the numerator of a fraction by any number, then, always make the fraction so many times less? Remember, then, that dividing the numerator divides the fraction. 16 3. What principle may be drawn from these two exercises? Ans. The fraction is {multiplied by multiplying? the numerator. divided dividing 4. If an apple be cut into two equal parts, what is one of them called? [See Chap. II., Section I.] How shall I write it on the black-board? Ans. By drawing a horizontal line, and writing 1 above and 2 below it? [Write it.] If I should divide one of these halves into 2 equal parts, what should one of them be called? [Write it.] Which of these fractions [1, 1 is the larger? How many times? Has multiplying the denominator by 2, then [point to the fractions on the blackboard], multiplied or divided the fraction? By what number? How many times is greater than ? Than? Than1⁄2? Will multiplying the denominator of a fraction by any number, then, always divide the fraction by that number? member, then, that multiplying the denominator divides the fraction. Re 5. [Write on the black-board.] If I divide the denominator 4 by 2, what will the fraction be? [Write it after the answer is received.] Which is the larger fraction? [Write .] If I divide the denominator of this fraction by 3, what will be the fraction? [Write it.] Which is the larger? How many times? Will dividing the denominator of a fraction, then, by any number, always multiply the fraction by that number? Remember, then, that dividing the denominator multiplies the fraction. 6. What principle have you discovered from the last two exercises? Ans. — The fraction is multiplied by dividing 4 16 { buttipiding the denominator. divided by 7. [Write on the board.] If this numerator [point] be multiplied by 2, or by any other number, what effect will be produced on the fraction? If this denominator be multiplied by 2, what will be the effect on the fraction? If a fraction be multiplied by 2, or by any other number, and then be divided by the same number, will the fraction be unchanged, or will it be greater or less than at first? What will be the effect, then, on this or any other fraction, if both terms be multiplied by the same number? Remember, then, that a fraction is unchanged when both terms are multiplied by the same number. 8. [Write on the board, as before.] If this numerator be divided by 2, or by any other number, what will be the effect on the fraction? If this denominator be divided by 2, or by any other number, what will be the effect on the fraction? If a fraction be divided by 2, or by any other number, and then be multiplied by the same number, will the fraction be unchanged, or will it be greater or less than at first? What will be the effect, then, on any fraction, if both terms be divided by the same number? Remember, then, that a fraction is unchanged, when both terms are divided by the same number. 9. What principle may be drawn from the last two exercises? Ans. — The value of a fraction is multiplying both terms by the or dividing S same number. not changed by [From this principle it is evident that the same fraction may be represented in literally an infinite variety of forms. For, by continually multiplying both terms of by 2, we have,, , &c., to infinity. And, what is still more remarkable, if the operation is commenced by the use of odd numbers as denominators, the fraction, or any other, may be represented in an infinite series of forms, each series of which may be continued to infinity. From these considerations, the extraordinary fact results, that an infinite number of men, say, as an approximation, every person now alive, and all that ever have existed, or ever will exist, might be employed from the present moment through all eternity, each person writing a series of fractions, all equal to one another, yet no two composed of exactly the same figures. And still further, the time in which this indefinite number of persons might have been thus employed, may be extended through the past eternity as well as through the eternity to come.] [As the three principles developed above are exceedingly important, it may be well, the more strongly to impress them on the mind of the pupil, to present the subject in another point of view, as follows:] 10. [Write on the black-board.] Which of these two numbers is the dividend? [See p. 67, 1. 24.] Which is the divisor? Where is the quotient [or quota, or share]? Ans. Both terms, namely, the fraction. If the dividend or thing to be divided, be increased [point to it], will the quotient, or share, or fraction, be increased or diminished? What will be the effect if the dividend be decreased? What will be the effect on the quotient, then, of multiplying the dividend by 2, 3, 4, or any other number? [See the first principle developed in this section.] 11. The divisor shows the number of parts into which the dividend is to be divided: if that be increased, then, will the quotient (or share, or fraction), be thereby increased or diminished? What will be the effect if the divisor be decreased? What will be the effect, then, of multiplying it by 2, 3, or any other number? Of dividing it by any number? [See the second principle developed in this section.] 12. If both dividend and divisor are multiplied by the same number, what will be the effect on the quotient (or fraction)? If both be divided by the same number, what will be the effect? [See the third principle developed in this section.] 13. [Write the following exercises on the board, and the answers, as fast as they are given by the class, as follows: of 1 =, &c.] What is of 3? Of 6 ? of 1? of 2? Of 2? What is of 1? Of 2 Now look 124 2 Of 3? Of 4? g of 1? at the board, and say, did you get these answers by adding, subtracting, multiplying, or dividing? What is of ? of 1 ? 1 of 1 ? of? What is the opération, then, when the numbers on each side of of are both fractions? What, then, does the word of imply, when connected with fractions? Remember, then, that the word of connected with fractions always implies multiplication. What is the divisor in ? In ? Perform the division in both these cases? Does the numerator, then, always express the fraction when the denominator is 1? Does the value of an integer, then, remain unchanged when 1 is placed under it as a denominator? 14. What principle is involved in the last exercise? Ans. A whole number may be expressed fractionally by writing 1 under it as a denominator. 15. Recapitulation. What effect is produced on a fraction by multiplying its numerator? By multiplying its denominator? By dividing its numerator? By dividing its denominator? By multiplying both its terms? By dividing both its terms? What does the word of imply when connected with a fraction? How may an integer be expressed fractionally? 16. What are the principles developed in this section? Ans. 1. If multiplication or division be performed on the numerator, the same effect is produced on the fraction. 2. If multiplication or division be performed on the denominator, a contrary effect is produced on the fraction. 3. No change of value is produced on the fraction when both terms are multiplied or divided by the same number. 4. A whole number may be expressed fractionally by writing 1 under it as denominator. 5. The word of connected with a fraction implies multiplica tion. |