28. A can do a certain piece of work in 8 days, and B can do the same in 6 days; in what time can both together do it? Ans. 3 days. 29. A merchant sold 5 barrels of flour for $321, which was f as much as he received for all he had left, at $1 a barrel; how many barrels in all did he sell ? Ans. 18. 30. What is the least number of gallons of wine, expressed by a whole number, that will exactly fill, without waste, bottles containing either 8, , 9, or 4 gallons ? Ans. 60. 31. A, B, and C start at the same point in the circumference of a circular island, and travel round it in the same direction. A makes of a revolution in a day, B 17, and C. In how many days will they all be together at the point of starting ? Ans. 1781 days. 32. Two men are 64 miles apart, and travel toward each other; when they meet one has traveled 53 miles more than the other; how far has each traveled ? Ans. One 294 miles, the other 353 miles. 33. There are two numbers whose sum is 1 jo, and whose difference is }; what are the numbers ? Ans. i and zu 34. A, B, and C own a ferry boat; A owns of the boat, and B owns is of the boat more than C. What shares do B and C own respectively? Ans. B, 4; C, &. 35. A schoolboy being asked how many dollars he had, replied, that if his money be multiplied by it, and of a dollar be added to the product, and of a dollar taken from the sum, this remainder divided by would be equal to the reciprocal of of a dollar. How much money had he? 36. If a certain number be increased by 18, this sum diminished by 3, this remainder multiplied by 5%, and this product divided by 12, the quotient will be 7}; what is the number? 37. If 3 of of 3 times 1, be multiplied by }, the product divided by 3, the quotient increased by 41, and the sum diminished by of itself, what will the remainder be? Ans. 6,805 Ans. 202. A Decimal Fraction is one or more of the decimal divisions of a unit. Notes.—1. The word decimal is derived from the Latin decem, which signiLes ten. 2. Decimal fractions are commonly called decimals. 203. In the formation of decimals, a simple unit is divided into ten equal parts, forming decimal units of the first order, or tenths, each tenth is divided into ten equal parts, forming decimal units of the second order, or hundredths; and so on, according to the following TABLE OF DECIMAL UNITS. 1 single unit equals 10 tenths; 10 hundredths; 10 thousandths; 1 thousandth 10 ten thousandths etc. etc. 204. In the notation of decimals it is not necessary to employ denominators as in common fractions; for, since the different orders of units are formed upon the decimal scale, the same law of local value as governs the notation of simple integral numbers, (57), enables us to indicate the relations of decimals by place or position. 205. The Decimal Sign (.) is always placed before decimal figures to distinguish them from integers. It is commonly called the decimal point. When placed between integers and decimals in the same number, is sometimes called the separatrix. 206. The law of local value, extended to decimal units, assigns the first place at the right of the decimal sign to tenths; the second, to hundredths; the third, to thousandths; and so on, as shown in the following Decimals. Integers. 207. The denominator of a decimal fraction, when expressed, is necessarily 10, 100, 1000, or some power of 10. By examining the table it will be seen, that the number of places in a decimal is equal to the number of ciphers required to express its denominator. Thus, tenths occupy the first place at the right of units, and the denominator of yhas one cipher; hundredths in the table extend two places from units, and the denominator of too has two ciphers; and so on. 208. A decimal is usually read as expressing a certain number of decimal units of the lowest order contained in the decimal. Thus, 5 tenths and 4 hundredths, or .54, may be read, fifty-four hundredths. For, + 16 = 36 209. From the foregoing explanations and illustrations we derive the following PRINCIPLES OF DECIMAL NOTATION AND NUMERATION. I. Decimals are governed by the same law of local value that governs the notation of integers. II. The different orders of decimal units decrease from left to right, and increase from right to left, in a tenfold ratio. III. The value of any decimal figure depends upon the place it occupies at the right of the decimal sign. IV. Prefixing a cipher to a decimal diminishes its value tenfold, since it removes every decimal figure one place to the right. V. Annexing a cipher to a decimal does not alter its value, since it does not change the place of any figure in the decimal. VI. The denominator of a decimal, when expressed, is the unit, 1, with as many ciphers annexed as there are places in the decimal. VII. To read a decimal requires two numerations; first, from units, to find the name of the denominator; second, towards units, to find the value of the numerator. 210. Having analyzed all the principles upon which the writing and reading of decimals depend, we will now present these principles in the form of rules. RULE FOR DECIMAL NOTATION. I. Write the decimal the same as a whole number, placing ciphers in the place of vacant orders, to give each significant figure its true local value. II. Place the decimal point before the first figure. RULE FOR DECIMAL NUMERATION. I. Numerate from the decimal point, to determine the denominator. II. Numerate towards the decimal point, to determine the numerator. III. Read the decimal as a whole number, giving it the name of its lowest decimal unit, or right hand figure. EXAMPLES FOR PRACTICE. Express the following decimals by figures, according to the decimal notation. 1. Five tenths. Ans. .5. 2. Thirty-six hundredths. Ans. .36. 3. Seventy-five ten-thousandths. Ans. .0075. / ا ا ا را 4. Four hundred ninety-six thousandths. 8. Four hundred thirty-seven thousand five hundred fortynine millionths. 9. Three million forty thousand ten ten-millionths. 12. Four hundred ninety-five million seven hundred five thousand forty-eight billionths. 13. Ninety-nine thousand nine ten-billionths. 14. Four million seven hundred thirty-five thousand nine hundred one hundred-millionths. 15. One trillionth. vite, 16. One trillion one billion one million one thousand one ten. trillionths. Y? 117/ 17. Eight hundred forty-one million five hundred sixty-three thousand four hundred thirty-six trillionths. 600647 5684 9.6 18. Nine quintillionths. ?11,6!! Ans. 46.4. 20. 105 26. 205-65 27. 60-36 22. 28. 705 25. 46,6 100 100000 5 1000 11 10000 85 100000 100004 1000000 ī000000000 29. 300-10001001 24. 30. 5270000000000 Read the following numbers : 31. .24. 38. 8.25. 32. .075. 39. 75.368. 33. .503. 40. 42.0637. 34. .00725. 41. 8.0074. 35. .40000004. 42. 30.4075. 36. .0000256. 43. 26.00005. 37. .0010075. 44. 100.00000001.. 704 10000000 23. 100000,000 |