explanation to make it intelligible to the pupil. We, there fore, prefer that here presented. The principle of this is clear and simple, and with it the pupil is already familiar (§ 71. 1, 2, 3, 4). 19579 58053 Ans. . Ans. . Ans. 1. Ans. 17. Ans. 123. (7.) Divide (8.) Divide of 1⁄2 by 3 of 1. (9.) Divide of of by 1 of of %. (10.) Divide 14 of ¦¦ by 3 of 9. (11.) of 23 less of 4 are how many times 34? Ans. 31. (12.) what result? Ans. 84. (13.) How much cloth, at 1 dollar (that is, dollars) a yard, can be bought for 4 dollars? Ans. 24 yards. (14.) A man distributed 82 bushels of wheat among some poor persons, giving 1 bushel to each; how many did he give it to? 5 persons. (15.) If a soldier is allowed 1 pound (that is, of a pound) of meat in a day, to how many soldiers would 6§ pounds be allowed? Ans. 4 soldiers. multiplied by 7 and the product divided by gives (16.) If 1 ton of hay will keep a horse through the winter, how many horses will 10 tons keep? Ans. 6 horses. (17.) At 2 dollars a box, how many boxes of raisins can be bought for 10 dollars? Ans. 4 boxes. Ans. 5 pounds. (18.) At 1 dollar a pound, how many pounds of indigo can be bought for 93 dollars? (19.) At 14 dollar a barrel, how many barrels of raisins can be bought for 93 dollars? (20.) At of a dollar a piece, how kin can be bought for 83 dollars? (21.) At § of a dollar a pound, how can be bought for 7 dollars? Ans. 6 barrels. many pieces of nan Ans. 10 pieces. many pounds of tea Ans. 10 pounds. DECIMAL FRACTIONS. $84. 1. In notation of whole numbers, it has been seen that the local value of figures varies according to the place in which they stand (§5, 3), that every figure designates a quantity ten times greater for every place it is removed to the left from that of simple units (7, 5). The increase from right to left was, therefore, tenfold (§7. 5). 2. Now, it is obvious, that if we wish to denote quantities less than unity, otherwise than by Vulgar Fractions, so that the notation of the numbers expresing them may correspond with that of whole numbers, we should write them in places below unity, that is to the right of unity, that being the lowest place in whole numbers, and make their decrease from unity to the right tenfold. 3. On such a system the first place to the right of unity would be tenths of units, the second place hundredths, the third thousandths, and so on. 4. Thus, there would be implied below each figure a denominator of 10, 100, 1000, or 1 with such a number of ciphers annexed as would denote the place occupied by the figure acording to the notation. 5. It would then be desired to separate by some mark or sign between the parts thus written and the whole numbers. This might be done by a simple point, or comma (,), which, from the end it would answer, might be called a decimal point, or separatrix. 6. Such a method of denoting parts we have; and differing from vulger fractions in having the denominator implied, not expressed, and this determined by place in notation; the notation being from left to right, by a fixed decrease, in tenths, or tenfold, it is called decimal fractions. 7. Therefore, a decimal fraction is a fraction which is expressed by writing the numbers denoting the parts at the right of a point, or comma, called a decimal point, or separating. The next place to the right would be Tens of Trillionths; the next, Hundreds of Trillionths, and so on. 9. The names used in Notation of whole numbers are used in succession from left to right, with only the termination changed by the addition of th, or ths, at the end. 10. To each of these figures there is implied a denominator, corresponding to the name of its place. 11. This denominator is always 1, 'with as many ciphers annexed as there are figures in the numerator. To the 4, at the left in the above, the denominator implied is 10; to the 6, 100; to the 4, 1000; and so on. 12. Decimals are read by calling each figure from left to right separately, naming the part which it denotes of its implied denominator; as, in the above, four tenths, six hundredths, one thousandth, and so on; or, by calling them off together as whole numbers, naming the implied denominator to the last figure, as the denominator to the whole; as four hundred and sixty-one Billions, five hundred and eighty-nine Millions, seven hundred and fifty-one Thousand, and thirty-one Trillionths; or, shorter, mentioning the (,) point, and then reading the subsequent numbers simply as they occur, leaving the denominators out, as understood from the principles of the system. 13. As in Vulgar Fractions we write a fraction to a whole number and form a mixed number (§ 56.), so in Decimals. Thus, 4,5; 89,785. 14. Examples. (1.) Seventy-six hundredths. (2.) Forty-nine thousand, four hundred, and nine tenths. (3.) Sixty-four trillions, and thirty-one millionths. (4.) Five hundred and seventy, and six tenths. (5.) Four hundred, and nine ten thousandths. (6.) Twenty, and two hundredths. (7.) Eighty-one hundred, and four ten thousandths. (10.) Eight hundred thousand, and twenty-five thousandths. (11.) Eleven, and seven billionths. (12.) Four hundred and twenty-one, and nineteen thousandths. 15. From the nature of decimals, we deduce four fundamental propositions. § 85. 1. PROPOSITION I. A cipher prefixed to a decimal decreases its value tenfold. 2. Illustration. To ,5 prefix 0. 05. Here the expression ,5==; but,05-18-30. 3. Explanation. By prefixing ciphers to a decimal, the figure or figures previously composing it are made to stand as many places farther to the right of the separatrix as there are ciphers prefixed; and, consequently, their value is decreased so many times tenfold (§ 84.-6). 4. Examples. (1.) Change ,5 to five thousandths. § 86. 1. PROPOSITION II. A cipher annexed to a decimal does not alter its value. 2. Illustration. To ,5 annex 0. ,50. Here the expression,5-1; and,50%=1. 3. Explanation. This operation corresponds with that of prefixing ciphers to a whole number, which does not vary the value of that whole number, because the increase being from right to left, and the places being so named, a place occupied by a cipher, which has no value (§ 4. 3), would be read as nothing of that place, or disregarded. So, here, the decrease being from left to right, the succeeding places can have significancy only as they are oc cupied by significant figures; for if, at the end, at the right, there stand a cipher, and it were named in connection with its place, it would be read as nothing of that place, or disregarded. 4. Examples. (1.) Change ,8 to eighty hundredths. (2.) Change,046 to four hundred and sixty thousandths. (3.) Change,74 to seventy-four thousand ten hundred thousandths. (4.) Change,9 to nine hundred thousandths. § 87. 1. PROPOSITION III. Removing the decimal point one place to the left, decreases the fraction tenfold; or divides it by 10. 2. Illustration. 43,752. The decimal point removed one place to the left, gives 4,3752, 3. Explanation. The explanation to Prop. I. suffices for this; for the principle, it is obvious, is the same in each; or, we may add, that the removing of the point to the left puts at its right in the place of decimals, and so makes decimals, so many of the whole numbers as by the removal of the point are brought to its right. 4. Examples. (1.) Diminish ,15 a thousandfold. |