CHAP. VI. FRACTIONS. SECTION 1. Note to Teachers. The subsequent progress of the learner, will depend much on a proper conception of the division of unity, and a correct application of the nomenclature of fractions. Therefore, this section, however simple it may appear, should not be slighted. It should be recited with the books closed. The picture of a board. This board, as it is presented above, is a whole thing. The same board appears hereafter divided into parts; and the parts are named according to their number and size. Divided now into 2 equal parts. One of these parts is one-half. 1. How many halves are there in the whole of any thing? 2. Suppose I can write a letter on 1-half of a sheet of paper; how much paper shall I use, in writing 2 letters ? 3. How much is i-half and 1-half, added together? 4. How many thirds are there in the whole of any thing? 5. If a carpenter can make 3 door-panels of i board, what part of one board will he use, in making 1 panel? 6. Which is the greater part, 1-half, or 1-third ? 7. How many fourths are there in the whole of 1 thing? 8. I gave 1-fourth of an orange to John, and 2-fourths to Frances. How much of the orange did I give away! 9. Which is the greater part, l-third, or 1-fourth? Divided now into 5 equal parts. One of these parts is one-fifth. 10. How many fifths are there in the whole of any thing? 11. Charles divided a melon, equally among 5 boys. What part of the melon, [how many fifths,] had 2 boys ? 12. Which is the smaller part, 1-fourth, or 1-fifth ? Divided now into 6 equal parts. 13. How many sixths are there in the whole of any thing? 14. If 3 girls and 2 boys should each of them eat 1-sixth of a pie, what part of the whole pie would they all eat? 15. Which is the greater part, i-fifth, or 1-sixth? Divided now into 7 equal parts. One of these parts is one-seventh. 16. How many sevenths are there in the whole of 1 thing? 17. John broke off 2-sevenths of a new pencil, and cut off 1-seventh more. How much of it was then wasted ? 13. Which is the smaller part, 1-sixth, or 1-seventh? Divided now into 8 equal parts. One of these parts is one-eighth. 19. How many eighths are there in the whole of 1 thing? 20. If a boy earn 3 eighths of one dollar, and find 4. eighths more, what part of one dollar will he then have? 21. Which is the smaller, 1-seventh, or l-eighth ? Divided now into 9 equal parts. One of these parts is one-ninth. 22. How many ninths are there in the whole of 1 thing? 23. Stephen paid 3-ninths of all his money for a slate, and 6-ninths for a blank-book. How much had he left? 24. Which is the greater part, 1-eighth, or 1-ninth? Divided now into 10 equal parts. One of these parts is one-tenth. 25. How many tenths are there in the whole of 1 thing? 26. If a book cost 5-tenths of a dollar, and a pepknife cost 4-tenths, what part of 1 dollar will they both cost ? 27. Which is the greater part, 1-ninth, or l-tenth? Remark 1st. It appears from the examples above, that, ONE-HALF of any thing, is one of two equal parts of the thing;-ONE-THIRD of any thing, is one of three equal parts of the thing; -ONE-FOURTH of any thing, is one of four equal parts of the thing; and so on. Remark 2nd. The greater the number of parts is, into which any thing is divided, the smaller the parts are. SECTION 2. Note to Teachers. One object in this section is, to lead the pupil to apply correctly the terms expressing fractional parts. Every answer, therefore, must be given in a vulgar fraction, unreduced. For example, two fourths is the answer which must be given to the 3d question. The books to be closed during the recitation of this section. 1. If we divide any thing into 2 equal parts, and take away 1 of the parts, how much of the thing is left ? 2. If we divide any thing into 3 equal parts, and take away 2 of the parts, how much of the thing is left? 3. If we divide any thing into 4 equal parts, and take away 2 of the parts, how much of the thing is left ? 4. If we divide any thing into 5 equal parts, and take away 3 of the parts, how much of the thing is left ? 5. If we divide any thing into 6 equal parts, and take away 3 of the parts, how much of the thing is left? 6. If we divide any thing into 7 equal parts, and take away 2 of the parts, how much of the thing is left? 7. If we divide any thing into 8 equal parts, and take away 5 of the parts, how much of the thing is left ? 8. If we divide any thing into 9 equal parts, and take away 3 of the parts, how much of the thing is left? 9. If we divide any thing into 10 equal parts, and take away 4 of the parts, how much of the thing is left ? 10. Into how many parts must any thing be divided, so that 1 part shall be 1-eleventh ? - Into how many, so that 1 part shall be l-twelfth ?-1-thirteenth ? SECTION 3. Note to Teachers. The learners may be referred to Remark lst. under the first section of examples in this chapter, for a correct form of expressio to be adopted in answering the 1st., 4th., 7th., and other similar questions in this section. Books to be closed during the recitation of this section. 1. What is meant by one-half of any thing? 2. Suppose you have 1-half of 1 dollar;—what part of a dollar more must you get, to make up 1 dollar? 3. How many halves are equal to a whole one? 4. What is meant by one-third of any thing? 5. If I should cut 1 orange into thirds, and give you 2-thirds of it, what part of an orange would you still want, to make up 1 orange by joining the parts together? 6. How many thirds are equal to a whole one ? 7. What is meant by one-fourth of any thing? 8. Suppose you have 1-fourth of 1 dollar,--what part of a dollar must you get, to make up 1 dollar ? 9. How many fourths are equal to a whole one? 10. What is meant by one-fifth of any thing ? 11. If I should cut 1 apple into fifths, and give you 4-fifths of it, what part of an apple would you still want, to make up 1 apple by joining the parts together? 12. How many fifths are equal to a whole ? 13. What is meant by one-sixth of any thing? 14. If I own 2-sixths of 1 acre of land, and I wish to own 1 acre, what part of 1 acre must I buy? 15. How many sixths are equal to a whole one ? 16 What is meant by one-seventh of any thing? 17. A man bought 4-sevenths of a pound of tea at one shop, and enough more at another shop to make 1 pound. What part of 1 pound did he buy at the last shop? 18. How many sevenths are equal to a whole one? 19. What is meant by one-eighth of any thing? 20. James had 5-eighths of a dollar given him, and he earned 3-eighths more. How much money had he then! 21. How many eighths are equal to a whole one? .22. What is meant by one-ninth of any thing? 23. If I have 7-ninths of 1 acre of land, and I wish to own 1 acre, what part of 1 acre must I buy? 24. How many ninths are equal to a whole one ? one 25. What is meant by 'one-tenth of any thing? 26. Suppose you have 8-tenths of 1 dollar, --what part of a dollar must you get, to make up 1 dollar? 27. How many tenths are equal to a whole one ? RELATIONS OF NUMBERS. SECTION 4. Note to Teachers. The object of this section is, to show the correspondence of the division of a unit, with the division of a collection of units. The ques. tions that inquire, what part of one number is another number, must be answered in the terms of vulgar fractions. For instance,-the answers required in the 9th example are,-1 is 1-fourth of 4; 2 is 2-fourths of 4; 3 is 3-fourths of 4. A collection of units is now to be viewed as a single thing; therefore the verb singular will be used thus-3 tiines 4 is 12. 1. If 1-half of a sheet of paper be worth 1 cent, what is a whole sheet worth? 2. Suppose 2 cents are lying upon the desk before us; -what part of the 2 cents is 1 cent? 3. What part of 2 is 1 ? 4. If 1-third of a loaf of bread be worth 1 cent, what is 2-thirds of it worth? What is a whole loaf worth? 5. Suppose 3 cents are in a pile before us;--what part of the pile is 1 cent? What part of the pile is 2 cents ? 6. What part of 3 is 1 ? What part of 3 is 2? 7. If 1-fourth of a yard of ribbon cost 1 cent, what will 2-fourths of a yard cost? What will 3-fourths of a yard cost? What will a whole yard cost? 8. Suppose 4 cents are in a pile before us;—what part of the pile is 1 cent?, is 2 cents ? is 3 cents ? 9. What part of 4 is 1? is 2? is 3 ? 10. If 1-fifth of a barrel of flour be worth 1 dollar, what is 2-fifths of a barrel worth? 3-fifths of a barrel 4-fifths of a barrel ? What is 1 barrel worth? 11. What part of 5 is 1 ? is 2? is 3? is 4 ? 12. If 1-sixth of a yard of ribbon cost 1 cent, what will 2-sixths of a yard cost? 3-sixths of a yard? 5-sixths of a yard? What will 1 yard cost ? 13. What part of 6 is 1 ? is 2? is 3? is 4? is 5 ? 14. If a horse trot 1 mile in 1-seventh of an hour, how many miles will he trot in 2-sevenths of an hour in 6-sevenths of an hour ? How many miles in 1 hour ? 15. What part of 7 is 1 ? is 2? 'is 3? is 4 ? is 6? |