2. What do you understand by one twentieth of any thing? 6 twentieths? 3. What do you understand by 4 fifths of any thing? 3 fifths? [The teacher should continue to ask questions similar to these, until the pupil answers without hesitation.] 4. 2 is 5. 2 is 6. 2 is 7. 2 is 8. 2 of 9. 1 is number? of what number ? 3 ? 4?5? 6? 7? 8? 9 ? 10? 11? of what number? 3 ?4?5? 6?7?8? 9? 10? 11 ? 12 ? of what number? 3 ?4?5? 6?7? 8? 9 ? 10? 11 ? 12? of what number? 3 ? 4? 5? 6? 7? 8 ? 9 ? 10? 11 ? 12? of what number? 3 ? 4?5? 6?7? 8? 9 ? 10 ? 11 ? 12? of what number? 1 is of what number? 2 is of what 10. 1 is of what number? 2 is of what number ? 3 is ‡ of what number? 6 11. 1 is of what number? 2 is & number? 12. 2 is of what number? 4 is number? of what number? 3 is 2 of what of what number? 6is of what 13. 4 is of what number? is of what number? 12 is of what number? 14. 6 is of what number? 9 is of what number? 12 is of what number? 15. 8 is of what number? 12 is of what number? 16 is of what number? § XXVII. Expressions like the above are called FRACTIONS. Then, FRACTIONS ARE EXPRESSIONS FOR PARTS OF NUMBERS. They are called fractions from a Latin word which means broken; because they stand for numbers divided or broken into parts. The term integer, a Latin word, signifying whole, is applied to the one whole thing or unit, of which fractions are broken parts. THE LOWER NUMBER IN A FRACTION IS CALLED THE DENOMINA. TOR; because from this, the Fraction receives its name, or denom. ination. THE UPPER NUMBER IN A FRACTION IS CALLED THE NUMERATOR ; because from this, we know the number of parts, for which the Fraction stands. The Numerator is, usually, less than the DenominaWhen this is the case, the Fraction is called a PROPER FRACTION. Sometimes, however, the Numerator is not less than the Denominator, but is equal to it, or greater.. When this is the case, the Fraction is called an IMPROPER FRACTION. tor. It has been seen, that, when the Numerator and Denominator are equal, the Fraction is equal to a whole one, or a unit, Thus, 2, 3, 4,,, &c. are each equal to 1. Then, as the Numerator of a Proper Fraction is never as great as its Denominator, A PROPER FRACTION IS ALWAYS LESS THAN A UNIT. Of course, AN IMPROPER FRACTION IS NEVER LESS THAN A UNIT. NUMBER. 5 is an Improper Fraction. It may be separated into the two Fractions and t. 4, as we have seen, is equal to 1. Therefore is equal to 1 and, or, as it is commonly written, 1. A whole number and a Fraction, written thus, are together called a MIXED 10 is an Improper Fraction, and may be separated into and. But each of these is equal to 1. Therefore, they are together equal to 2. is more than . Hence, 2}. Ilence it appears, that, as often as the Denominator of an Improp er Fraction is contained in the Numerator, so many whole ones, or Integers are contained in the Fraction; and, that if the Denominator will not divide the Numerator exactly, a Proper Fraction will re main. Hence, also, any Improper Fraction may be changed, or reduced to a whole or mixed number; and in order thus to reduce it, we must, DIVIDE THE NUMERATOR BY THE DENOMINATOR; WRITE THE REMAINDER, IF THERE BE ANY, OVER THE DENOMINATOR, AND ANNEX THE FRACTION, THUS FORMED, TO THE QUOTIENT. EXAMPLES FOR PRACTICE. 1. Reduce to a whole or mixed number. Ans. 93. 2. Reduce. Ans. 93. 5. Ans. 94. 7. Ans. 15}. 19. Ans. 23. 8 3. Reduce 15. Ans. 523. 2975. Ans. 565. 21§25. Ans. 2425. 3 7 4. Reduce 121, 6276. 5184 32. 915873 5. 3 6. Reduce 7112345499. 59248321768, 8 9 T༠༠༠,༠༠༠7. 495 635 0217 600304002 6 3 3 2 2 2 1 1 136 6 In the following examples, this process is reversed. 1. How many 4ths. in 1? In 13? 2 2. How many 5ths. in 1? How many in 14? In 12? In 5? In 1? In 13 ? In 7? 3. How many 7ths. in 7? In 8? In 12? In 73 ? In 59 ? 4. How many 12ths. in 9? In 7? In 3 ? In 5 ? In 8 7? 5 5. How many 6ths. in 3? In 4? S? In 94? In 12? 6. How many 27ths. in 3? In 2? 7. How many 19ths. in 15? In 13% In 1718? Here we multiply the whole number by the number expressing the parts, and add in the additional parts, if there be any. By this process, we obtain an Improper Fraction. Hence, the process is called, reducing a whole or mixed number to an Improper Fraction. The operation, as performed above, is as follows: MULTIPLY THE WHOLE NUMBER, BY THE NUMBER EXPRESSING THE PARTS, AND ADD THE NUMERATOR OF THE FRACTION, IF THERE BE ANY, TO THE PRODUCT. THE SUM WILL BE THE NUMERATOR AND THE MULTIPLIER, THE DENOMINATOR OF THE RESULTING IMPROPER FRACTION. § XXVIII. From the nature of Fractions, it is evident that Division may be expressed by a Fraction. For, in Division, the dividend is to be separated into a number of parts, denoted by the divisor. Hence, DIVISION MAY BE INDICATED BY WRITING THE DIVISOR UNDER THE DIVIDEND IN THE FORM OF A FRACTION. This mode of indicating Division was given § xxv. ALL FRACTIONS ARE INSTANCES OF DIVISION, IN WHICH THE NUMERATOR IS THE DIVIDEND, AND THE DENOMINATOR THE DIVISOR. Of course, THE VALUE OF A FRACTION IS THE QUOTIENT, WHICH ARISES FROM DIVIDING THE NUMERATOR BY THE DENOMINATOR. When the dividend is not less than the divisor, division may be both indicated and performed. When it is less however, Division can only be indicated. We may therefore give a brief rule, for the case when the dividend is less than the divisor. We are the rath er inclined to give it, though the case is very simple, because we have often seen pupils much perplexed by it. If the dividend be less than the divisor, WRITE THE DIVISOR UNDER THE DIVIDEND, AND THE FRACTION FORMED WILL BE THE answer. EXAMPLES FOR PRACTICE. Divide 37 by 45. 81 by 83. 97 by 120. 15 by 31. 17 by 19. 27 by 245. 383 by 384. 2 by 231. 3 by 756. 16 by 165. 4 by 71. 326 by 525. 482 by 491. 374 by 1,693. The pupil will observe, that where a less number is divided by a greater, the fractional quotient found, shows what part the dividend is ofthe divisor. Thus, 1 is the seventh part of 7, and 1÷7=4. 2 is two sevenths of 7 and 2÷7=4, and so on. It is common, likewise to extend the term part, to numbers, greater than those, of which they are said to be parts; thus. When the *question is asked what part of 3 is 1, the answer is, one third,=. When it is asked what part of 3 is 2, we say two thirds,=. In like manner, when it is asked, what part of 3 is 4, the answer is four thirds, . When it is asked what part of 3 is 5, the answer is five thirds, 5, and so on. Hence, Every Fraction shows what part the Numerator is of the Denominator. Of course, INDICATING DIVISION IS FINDING WHAT PART OF THE DIVISOR IS EQUAL TO THE DIVIDEND. Therefore, to find what part one number is of another, MAKE THЕ NUMBER CALLED THE PART THE NUMERATOR OF A FRACTION, AND THE OTHER NUMBER THE DENOMINATtor. NOTE. This rule must be strictly followed, without regarding which number is the greater. This is sometimes called, finding the RATIO of one number to another. (§ xc.) The Fraction obtained expresses the RATIO of the denominator to the numerator. EXAMPLES OF FINDING PARTS, OR RATIOS. 1. What part of 5 is 1? A. 3. 2. What part of 9 is 6? A. §. What part is 12 of 5? A. 12 3. What part is 16 of 25? What part of 25 is 16? 4. What part of 3,684 is 27,942? What part is 3,684 of 27,942 ? § XXIX. 1. A Gentleman paid 6,372 dollars, for 36 acres of land. How much did he give an acre? Here our divisor consists of two figures. We cannot therefore, divide by the last rule. But we know that 4 times 9 are 36; and that 4 times 9 acres are 36 acres. If we divide, then, by 4, we obtain the price of 9 acres. We can then divide this price by 9 and obtain the price of 1 acre. A. 4)6372 9)1593 177 2)2016 4)1008 2. If 56 hogsheads of molasses cost 2,016 dollars, what cost 1 hogshead? 2X4X7-56. Therefore, if we divide by 2, we shall obtain the price of half as many hogsheads, that is, of 28 hogsheads. If * we divide this result by 4, we shall obtain the price of 6 hogsheads. If we divide this result by 7, we shall obtain the price of 1 hogshead, which is required. Hence, to divide by a composite number, RESOLVE THE DIVISOR 7)252 A.. 36 INTO FACTORS, AND DIVIDE BY THOSE FACTORS SUCCESSIVELY. EXAMPLES FOR PRACTICE. 3. At 15 dollars a hhd., how many hhds. of sugar can be bought for $4,500? A. 300. 4. At $18 a ton, how many tons of iron can be bought for $2,250 ? A. 125. 5. If an acre of ground cost $25, what number of acres will $9,375 buy? A. 375. 6. If a hhd. of molasses cost $27, what number of hhds. will $5,940 buy? A. 220. 7. Divide 6,894 by 18 | 12. Divide Divide 9. 66 39,942 18 | 14. 66 273,045 by 15 714,357 by 21 2,295,495 66 45 3,575,635" 35 11. "27,966,232 265,824" 24 15. (6 333,225" 25 16. 66 56 It is to be observed, that, in many instances, we have remainders when the division is completed. The pupil may not readily discover, in dividing by composite numbers, how to find the true remainder. We will endeavour to explain the manner, by an example. 17. How many cubic yards in 369 cubic feet? There are 27 cubic feet in a cubic yard. Therefore, we must divide by 279X3. Divide first by 9 thus, 9)369 3)41 13+2 Rem. For 41, the first quoThen each unit of the quoNow, in dividing this 41, There is no remainder-Then by 3; thus, Here we have 2 remainder-But this does not signify that there are only 2 cubic feet left. tient shows that there are 41 9's in 369. tient 41, is 9 units of the dividend, 369. there is left a remainder of 2 units. But if 1 of these units is 9 units of the dividend, 2 of them will be twice as much; that is, 18 of the dividend. Of course there are 18 cubic feet left. Then, when there is a remainder on the last Division, we must multiply that remainder, by the first divisor, to find the true remainder. 18. At 36 dollars a ton, how many tons of iron can I buy with $762? 35=7X5 Divide by 7, thus, 7)762 Multiply the last remainder 3 by the first divisor 7, as directed by rule. 7X3=21. Then if there had been no remainder on the first Division, 21 would be the true remainder. But the first division left 6, which must, therefore, be added to the 21, making 27 the true remainder. Then when the divisor is resolved into two factors, in order to find the true remainder, we must multiply the last remainder by the first divisor and add in the first remainder. For similar reasons, when the divisor is resolved into several factors, multiply each remainder, arising after the first division, by all the preceding divisors, and add the products to the first remainder. 19. At $25 a barrel, how many barrels of brandy can I buy for $263? A. 10 13. 20. At $27 an acre, how many acres of land can I buy for $988 ? A. 364. 21. Divide 853 by 54. 22. Divide 971 by 63. 23. Divide 4,761 24. § XXX. 66 9,893 A. 15 42. 1. At 10 cents a pound, how many lbs. of raisins can I buy for 80 cents? We saw in Multiplication, that, if a figure were removed one place to the left, it was increased in a ten-fold proportion; that is, it was multiplied by 10. |