SECTION IV. FRACTIONS. 21. If any number, or particular thing, be divided into two equal parts, those parts are called halves, if into 3 equal parts they are called thirds, if into 4 equal parts they are called 'fourths, or quarters, (11); and, generally, the parts are named from the number of parts into which the thing, or whole, is divided. If any thing be divided into 5 equal parts the parts are called fifths, if into 6, they are called sixths, if into 7, they are called sevenths, and so on. These broken, or divided quantities are called fractions. Now if an apple be divided into five equal parts the value of one of those parts would be one fifth of the apple, and the value of two parts two fifths, of the apple, and so on. Thus we see that the name of the fraction shows, at the same time, the number of parts into which the thing, or whole, is divided, and how many of those parts are taken, or signified by the fraction. Suppose I wished to give a person two fifths, of a dollar; I must first divide the dollar into 5 equal parts, and then give the person two of these parts. A dollar is 100 cents-100 cents divided into 5 equal parts, each of those parts would be 20 cents. Hence one fifth of 100 cents, or a dollar, is 20 cents, and two fifths, twice 20, or 40 cents. 22. The tediousness and inconvenience of writing fractions in words has led to the invention of an abridged method of expressing them by figures. One half is written, one third,, two thirds, 3, &c. The figure below the line shows the number of parts into which the thing, or whole, is divided, and the figure above the line shows how many of those parts are signified by the fraction. The number below the line gives name to the fraction, and is therefore called the denominator; thus, if the number below the line be 3 the parts signified are thirds, if 4, fourths, if 5, fifths, and so on. The number written above the line is called the numerator, because it enumerates the parts of the denominator signified by the fraction. As there are no limits to the number of parts into which a thing, or whole, may be divided, it is evident that it is possible for every number to be a numerator, or a denominator of a fraction. Hence the variety of fractions must be unlimited.. The following table is designed to familiarize the pupil with the method of writing fractions. TABLE OF FRACTIONS. The above table contains only a few of the most simple and common fractions; but enough, probably, to make the pupil familiar with the notation of fractions generally. While learning the notation and numeration of fractions, the pupils should be frequently exercised in writing fractions from dictation, upon slates, or a black board. In this way they may be very soon taught to express correctly, and with fa cility any fraction proposed. Note -The numerator is sometimes written before the denominator, as 1-2, or 3.8ths. 23. 1. If you can buy a barrellwhat willyard cost? 1 yard?1? of fish for 4 dollars, how much car 2? 23? you buy for 2 dollars? for 1 do!- When a whole number and a lar? for 3 dollars? for 5? for 6? 7 fraction are written together, as 1, 8?9? 10? 11 ? 12? or 2, they are called a mixed number. 2. What is meant by? by? 4 ? ? ? ? &c (11.) 3. What part of 2 is 1? Ans. 4. What part of 2 is 2? , or the whole of 2. 13. If of a yard cost 2 dollars, what will of a yard cost? 1 yard? A.1?2? When the numerator and denominator are just equal, it shows that there are as many parts in the fraction as the unit is divided in 14 What is a mixed number ! 15. If a bushel of wheat cost 56 cents, what will half a bushel cost??? 16 What does the denominator to, and the value of the fraction is of a traction signify? therefore one whole, or 1. 17. What does the numerator 5. What part of 3 is 1? is 2? 3 ? signify? 6. What part of 4 is 1? is 2? 3? 18 How would you proceed to 7. What part of 5 is 1? is 2? 3?give a person of an apple ? ? } ? 4? 5? 8. What part of 6 is 1? is 2? 3? 4? 5? 6? 19. How to give a person of a dollar? how many cents would you give him? 20. When a fraction is written, 9. What part of 7 is 1? is 2?3? 10. What is the value of why is the number below the line ? ? ? ? ? ? ? ? &c. called the denominator? 21. Why is the one above it 11. At 4 shillings a bushel, how called the numerator? much corn can be bought for 5 22. At 16 cents a pound, what shillings? 6?7? 8 ? 9 ? 10. will 34 pounds of loaf sugar cost? 12. If of a yard cost 1 shilling, 24. 1. Three are how many buy for 1 dollar? 2? 3? 4? 5? 6? times 2? Ans or once and 9. At 10 dollars a hundred weight, When the numerator of a fraction what part of a hundred can you is larger than the denominator, it buy for 1 dollar? for 3? 5?7? is called an improper fraction, and 10. In 11, how many times 2? may be changed into a whole, or mixed number by dividing the nu11. In 11, how many times 23? merator by the denominator. Ans., or 11 times. 2 Four are how many times 2?3? 4? 3. Five are how many times 4? 3? 2? 1? 4. Five are how many times 7? Ans., or 5 times .. 3?7? 5? 6? 12. In 12, how many times 4? 5? 9 7 13? 19? 13. In 13, how many times 6? 5? 4? 9? 13? 17? 14 In 15, how many times 5.? 4? 7? 10? 20? The number of times one num15. If you pay 16 cents for 12 ber is contained in another is call-apples how much is that a piece? ed the ratio of one to the other. 16. Divide 21 apples equally a5. If wheat be worth 8 shillings mong 9 boys, how many will each a bushel, what is of a bushel receive? worth??? ? 17. At 12 cents a pound, how 6. If strawberries be worth 8 much sugar can you buy for 18 cents a quart, how many can you cents? 15? 22? 27? 1? 8?3? buy for 9 cents? Ans. 1 qt. 7. At 8 cents a quart, how ry can you buy for 10 cents? 8. When hay is worth 9 dollars a ton, what part of a ton can you' 18 At 13 dollars a ton, how much ma-hay can you buy for I dollar? 7? 12?3? 17? 26? 30? 19? 28? 40? 19. At 9 cents a pound, what will half a pound cost? 25. I. How many thirds make placing under the sum the common a whole one? how many 4ths denominator. 5ths? 7ths? 6ths? 9ths? 10ths? 7. Wha is the sum of 2 ninths of 2. What part of an apple is of a dollar and 3 ninths of a dollar? an apple and of an apple? Ans. In this case we should first di3 of an apple. vide the dollar into 9 equal parts, When two or more fractions which would be 9ths, then take 2 have the same number for a de-parts, which would be 2-9ths, and nominator they are said to have a3 which would be 3-9ths, and adcommon denominator. ding them together their sum would be 5-9ths. 3. What part of an orange is of an orange and of an orange? Ans.or 8. What is the sum of 6-IIths and 3-IIths? 4-9ths and 3-9ths? of a 7-13ths and 5-13ths? 1-19th and 4 19ths? 12-25ths and 12-25ths? 4. What part of a dollar is dollar and? and ? and ? 5. What part of a foot is foot and and §? § and 6. What is the sun of of and ? of § and ? a of ?, and ?| 9. What part of a mile are 2-7ths of a mile 3-7ths and 1-7th of a mite? 10. What is the sum of of a From these examples the pupillyard 2-8ths and ? will perceive that fractions, having II. What is the sum of I-10th, 2 the same denominator, are added, 10ths and 3-10ths? I-11th, 2-11ths, by adding their numerators, and 3-IIths and 4-IIths? 12. If a guinea be 28 shillings, 4ths of the farm and 6-14ths? 5 what part of a guinea are 4 shil 17ths & 3-17ths? 2-11ths & 5-11ths ? lings and 7 shillings? 15. What part of an hour are 2 13. A week is 7 days; what part15ths of an hour and 7-15ths? 8 of a week are 4 days and 2 days 15ths & 6-15ths? 5-15ths & 4-15ths? 3 days and 2 days? 16. What is the sum of I-13th, 14 What part of a farm are 72-13ths, 3-13ths, 4-13ths & 2-13ths? 3. What is the difference be- II. A man having a pound of tween I fifth of a dollar and 3 fifths tea, lost of it, how much had he of a dollar? between 2 sixths and left? 5 sixths? 12 There is a post I fifth of 4. Take 3 sevenths of an orange which is in the mud, 2 fifths in the from 5 sevenths, what remains? water, and the rest above; how 5. From I apple take of an much is there above the water? apple, what remains? 13. A owns 4 twelfths of a prize, 6. From 1 pear take of a pear, B 6 twelfths, and C the remainder, what remains? what is C's share? We must first divide the pear in- 14. What part of a guinea is the to 3ds and then, taking away 2 of difference between 4 shillings and 7 shillings? Aus. 3 twenty-eighths them, there will be 1-3d left. 7. From I dollar take § of a dol- of a guinea. lar, what remains?" 8. From of a prize take what remains? 15 An eagle is 10 dollars, what part of an eagle is the difference between 3 dollars and 9 dollars? From these examples it will be 16. What part of a week is the seen that the difference between difference between I day and 7 two fractions whose denominators/days? are equal, is the difference of the 17. What part of numerators placed over the com-difference between mon denominator. of a foot? a foot is the of a foot and 9. Two men drew 9 thirteenths 18. What part of a mile is the of a prize; one man's share was 4 difference between 19 28ths of a thirteenths, what was the others mile, and 7-28ths of a mile? sheet cost? 2 sheets? 3 sheets? pound, what will 2 pounds of loaf of 12? 3 sugar cost? 3? 4?5?6? 17. If of the length of a pole 12. If 5 barrels of cider cost 15 be 3 feet, what is the whole length dollars, what part of 15 dollars willlot the pole? one barrel cost? 2 barrels ? 3 ? 4? 18. If of a guinea be worth 13. What is I-5th of 15? 2-5th of shillings, what is the worth of a 15? 3-5ths of 15? 4-5ths of 15? 14. If I yard of cloth cost of a dollar, what will 3 yards cost? and 3 and ? 3 whole guinea? 19. If one fifth of a crown be worth 22 cents, what is the worth of a whole crown? 15. If one orange cost of a 20. If of a bushel be 4 quarts, shilling, what will 2 oranges cost how many quarts are there in a 3?4?5?6?7?8? bushel? 16. At one fifth of a dollar a 28. 1. If I have 9 apples and had he left? what part of the divide of them equally between whole would that be? 3 boys how many do I give them 9. Henry had 15 cents and spent apiece? of them how many did he spend? 2. f 9 are how many times 3? how many bad he left? what part 3. A man having 12 hundred of the whole ? acres of land reserved 1-6th of it 10. A boy had 30 cents and paid for himself and divided the rest of 5 6ths of them for 5 oranges, how it equally between 5 sons, how much were they apiece? much did each receive? 11. A lad had 45 pears and sold 4. 5-6ths of 12 are how many 2-5ths of them for 27 cents, how times 2? much was that apiece? 5 of 16 are how many 2? 3? 4? times 12. Charles had 36 plums 7-9ths of which he divided among 4 of 6. 4-7ths of 21 are how many his playmates, how many did each times 6? 4? 3? 2? receive? how many had Charles 7. 5-9ths of 27 are how many left? what part of the whole ? times 7? 5? 10? 3? 2? 13. If 3 men will do a piece of 8. George had six apples and work in 12 days in what time will gave of them to Henry, how ma-2 men do it? ny did he give him? how many) 29. I. Which is most of an Why? Ans. Because is equal apple or § of an apple? Ans. of to 2-4ths and 2-4ths and are 4. an apple. 6. How many 6ths of an apple is 2. Why? Ans. Because the of an apple? of an apple? greater the number of parts a thing Aus. 2-6tbs, and 4-6ths. is divided into the less those parts, 7 How many eighths of a dolor because it takes 3 thirds, but on-lar is of a dollar? of a dollar? ly 2 halves to make an apple of a dollar? 3. Which is most or I-6th? 8. How many 10ths of a dollar I-5th or or I-14th? is of a dollar?of a dollar? 4. Why is more than I-6th? of a dollar? 1-5th of a dollar? 5. What part of an apple is and 3-5ths of a dollar? 4-8ths of a dol of an apple? Ans. of an ap-lar? ple. 19, What part of a dollar is of |