10. 5 of 72 how many ? of 72 are 4 of what number? s of 72 are 4 of how many times 9 ? Explain. 11. of 15 how many ? of 15 are o of what number? of 15 are for of how many thirds of 21 ? Explain. Form of the solution. - 1 of 15 is 3, and of course 12; if 12 be 196, I will be 2, and 1: 20; s of 21 is 7, and 20 is two times 7 and 4 of 7. 12. of 18 how many? of 18 are of what number? of 18 are g of how many sevenths of 35? Explain. 13. & of 21 how many ? of 21 are of what number? of 21 are of how many sixths of 24 ? Explain. 14. of 24 how many ? of 24 are 10 of what number? of 24 are 40 of how many fifths of 40 ? Explain. 15. { of 32 how many ? ã of 32 are of what number? 5 of 32 are of how many fifths of 35 ? Explain. 16. of 63 how many ? of 63 are of what number? of 63 are of how many ninths of 45 ? Explain. 17. 4 of 56 how many ? of 56 are of what number? of 56 are of how many fourths of 28 ? Explain. 18. 3 of 64 how many ? of 64 are 18 of what number? of 64 are fo of how many sixths of 30 ? Explain. 19. g of 72 how many ?' of 72 are id of what number? s of 72 are is of how many fifths of 40 ? Explain. SECTION X.- Practical Questions. 1. As 2 boys were counting their money, one said he had 20 cents. The other said, of your money is exactly of mine. How many cents had he? of 20=of what? 2. A merchant kept his silver money in one part of his till, and his cents in the other. Having found 18 dollars in the former, his clerk, who counted the cents, told him that of the dollars were equal in number to 1 of the cents. How many cents were in the drawer ? Prove. 3. A man being asked the age of his eldest son, replied; that his youngest son was 6 years old, and that is of the age of the youngest was just 1 of that of the eldest. What was the age of the eldest? Prove. 4. A man, being asked how many sheep he had, said, that he kept them in two pastures. In one pasture he had 24; and 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, s 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 18 of what were in his other pasture. How many had he in both pastures? Explain the process, or prove. 7. A farmer had sheep in three different pastures. In the first he had 100. In the second he had of 1 of what he had in the first. [How many are 1 of 100? Then how many are of that?] In the third he had % of of what he had in the second. How many sheep had he in all ? Explain the process. 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 şof 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 T = 1. Which number is the numerator of a fraction? Which the denominator? What are both called ? How much is 2 times it? How many times does it contain it? Ans. Two times. How much is 3 times it? How many times does & contain 16 ? How much is 5 times It? How many times does it contain to? [Point to the black-board.] Will multiplying 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. 2. Divide to by 2; in other words, what is the half of 12 ? How many times is fc contained in t? What is the third part of it? in other words, divide t? by 3. How many times is to contained in 1?? [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. 3. What principle may be drawn from these two exercises ? Ans. – The fraction is { multiplied by multiplying} the numerator. 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, Ans. -The fraction is divided by multiplying the denominator. 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 į greater than ? Than f? Than t'z? Will multiplying the denominator of a fraction by any number, then, always divide the fraction by that number ? Remember, then, that multiplying the denominator divides the fraction. 5. [Write on the black-board.] If I divide the denominator by 2, what will the fraction be? [Write it after the answer is received.) Which is the larger fraction ? [Write 5.] 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 ? multiplied by dividing? 7. [Write is 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 to 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 S multiplying both terms by the not changed by or dividing) same number. [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 1 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.] |