D 9 tons at 12 dollars, and the rest of his purchase to E at 13 dollars per ton; what did he gain on the hay? Ans. 83 dollars. 42. If 3 men can plough 15 acres of ground in 4 days, how many acres ought 5 men to plough in 7 days? Ans. 471 acres. 43. If a railroad car can run 35 miles in 3 hours, and 41 miles in 4 hours, and 62 its average rate per hour? miles in 5 hours; what will be Ans. 11 miles. 44. A bought of B 783 lars, and sold 500 bushels of it to C for 5621 dollars. much did he gain or lose per bushel on what he sold? bushels of wheat for $587 dol How Ans. Gained of a dollar. 45. Bought 175 cords of wood for 437 dollars, and sold 93 cords of it at a profit of half a dollar per cord. At what rate must the remainder be sold, to gain 973 dollars on the whole ? Ans. 31 dollars per cord. 46. A person who has a journey of 570 miles to perform, proceeds for 9 days at the rate of 334 miles per day. How much must his daily rate be increased or diminished, to complete the journey in 9 more days? Ans. Diminished 33 miles. 47. A person bought 19 barrels of apples, at 21 dollars per barrel. Having sold 121 barrels of them at 2 dollars a barrel, at what price per barrel must he sell the remainder, to gain 5 dollars on the whole? Ans. 217 dollars. 48. A purchased of B 40 yards of cloth for 260 dollars. He then sold to C of his purchase at a profit of of a dollar per yard, and the remainder to D, at a loss of of a dollar per yard; what did A gain or lose by these several transactions? Ans. Gained 7 dollars. CHAPTER V. DECIMAL FRACTIONS.-DECIMAL OR FEDERAL MONEY. DECIMAL FRACTIONS. (81.) A DECIMAL FRACTION is a number of tenths, hundredths, or thousandths, &c., denoted by one or more figures on the right of units, and after a point (.) which distinguishes them from integers (62). · Thus .3, three tenths; .35, thirty-five hundredths. The 1st figure after the decimal point denotes tenths, the 2d hundredths, the 3d thousandths, and so on; but they may all together be expressed in the denomination of the right hand figure. Thus .35 denotes 3 tenths and 5 hundredths, or 35 hundredths. What does .1, decimal point, 1, denote? What does .12 denote? What does .234 denote? .3546? .05? .006? .067? .0009? The simple term decimal is often used to designate a decimal Fraction. A Vulgar Fraction is one which is denoted by a numerator and denominator; as,, 7, 80.. 3 4 23 Scale of Decimals. (82.) In Decimals, as in integers, ten of any lower order make one of the next higher order; or one of a higher order makes ten of the next lower order. Thus 10 thousandths make 1 hundredth; 10 hundredths make 1 tenth. 2 units are how many tenths? 3 tenths are how many hundredths? 5 hundredths are how many thousandths? 40 thousandths are how many hundredths? From the preceding it follows, that (83.) Each 0 between the (.) and the first significant decimal figure, diminishes the decimal to one tenth of its value without the 0 Thus .03 is one tenth of .3. How may the figure 1 be made to denote 1 tenth? How may it be made to denote 1 hundredth? 1 thousandth? How may the figure 5 be made to denote 5 hundredths? 5 thousandths? O's annexed to decimals do not alter the values of the decimals; thus .1.10.100.1000, &c. Mixed Decimals. (84.) A mixed Decimal is a decimal fraction with a vulgar fraction annexed to denote a part of 1 tenth, or 1 hundredth, &c. .5 denotes 5 tenths, that is, 5 tenths and of 1 tenth. .253 denotes 253 hundredths, or 25 hundredths and of 1 hundredth. What does .3 denote ? .041? .123? .00051 ? Notation of Decimals. RULE XVIII. (85.) To denote, decimally, a Number of tenths, hun dredths, or thousandths, &c. Prefix the (.) to the number, with O's interposed, if necessary, to put the last figure in the given denomination, when the successive figures are called tenths, hundredths, thou sandths, ten-thousandths, &c., from the (.) towards the right. EXAMPLES. 1. To denote, decimally, 54 ten-thousandths. .0054. The () and 00 must be prefixed to 54, to put the 4 in the donomination of ten-thousandths, when the successive figures are called tenths, hundredths, &c., from the (.) towards the right. 2. To denote, decimally, 125 and 7 hundredths. 125.07. The () and 0 prefixed to 7 make the 7 denote 7 hundredths, to be placed on the right of the integral number 125. EXERCISES. Write in decimal figures each of the following Fractions. 1. Fifteen hundredths. 2. Nineteen thousandths. 7. One hundred thousandths. 8. Ten ten-millionths. 9. Forty-nine hundredths. 10. Seventeen ten-thousandths 11. Fifty-two thousandths. 12. Eight hund. thousandths. Write in integers and decimals each of the following Mixed Numbers-observing that, in the verbal expression, the comma (,) separates the fraction from the integer. 13. Four thousand and nine, and five thousandths. 14. Fifty-four thousand, three hundred and two thousandths. 15. Six hundred and twenty, and twelve hundredths. 16. Nine hundred and one, and five hundred and one millionths. 17. One million, and four thousand three hundred and ten hundred-thousandths. 18. Twenty thousand and seventeen, and nineteen tenthousandths. 19. Forty-seven thousand, and two hundred and twentyone thousandths. 20. Five millions two hundred and one thousand, and three tenths. 21. Seven hundred millions, and three hundred and nine thousandths. The principles of the preceding Rule will also enable the pupil to read any decimal fraction.-To prevent ambiguity in the enunciation of Mixed Numbers it will sometimes be expedient to insert the word decimal before the fraction; thus 300.005, three hundred and decimal 5 thousandths. 500.0002, five hundred and decimal 2 ten-thousandths. RULE XIX. (86.) To reduce a Decimal to a Vulgar Fraction. 1. Remove the (.), and under the given number of tenths, or hundredths, or thousandths, &c., set the proper denominator 10, or 100, or 1000, &c. 2. The Vulgar Fraction thus formed may often be reduced to lower terms. EXAMPLE. To reduce .125 to a vulgar fraction. .125=125 =}. 25 The given decimal being 125 thousandths, we take 1000 for a denominator, and reduce the 12 to its lowest terms (67). |