many children, grand-children, and great grand children are there? Ans. 32. 11. The distance around the earth is computed to be about 25000 miles : how long would it take a man to travel it, supposing him to travel at the rate of 35 miles a day? Ans. 71441 days. 12. The earth moves around the sun at the rate of 68 thousand miles an hour : how many miles does it travel in a day, and how many in a year? 1632000 in a day. Ans. { 595680000 in a year. 13. A farmer purchased a farm for which he paid 18050 dollars. He sold 50 acres for 60 dollars an acre, and the remainder stood him in 50 dollars per How much land did he purchase ? Ans. 351 acres. acre. $ 56. The unit 1 represents an entire thing; as I apple, 1 chair, 1 pound of tea. If we suppose one thing, as one apple, or one pound of tea, to be divided into two equal parts, each part is called one half of the thing. If the unit be divided into 3 equal parts, each part is called one third. If the unit be divided in 4 equal parts, each part is called one fourth. If the unit be divided into 12 equal parts, each part is called one twelfth ; and when it is divided into any number of equal parts, we have a similar ansion for each of the parts. equal parts of a thing are expressed thus: is read one half. one eighth. one fifteenth. one fiftieth. The 1, 1, &c. are called fractions. $ 57. Each fraction is made up of two numbers ; the number which is written above the line, is called the numerator ; and the one below it is called the denominator because it gives a denomination or name to the fraction. For example, in the fraction }, 1 is the numerator, and 2 the denominator. In the fraction }, 1 is the numerator, and 3 the denominator. The denominator in every fraction shows into how many equal parts the unit or single thing, is divided. For example, in the fraction }, the unit is divided into 2 equal parts; in the fraction }, it is divided into 3 equal parts; in the fraction t, it is divided into 4 equal parts, &c. In each of the fractions one of the equal parts is expressed. But suppose it were required to express 2 of the equal parts, as 2 halves, 2 thirds, 2 fourths, &c. We should then write, f they are read two halves. two thirds. two fifths, &c. If it were required to express three of the equal parts, we should place 3 in the numerator; and generally, the numerator shows how many of the er parts are expressed in the fraction. . For example, three eighths are written, four ninths. modo Ø 58. When the numerator and denominator are equal, the numerator expresses all the equal parts into which the unit has been divided : therefore, the value of the fraction is equal to 1. But if we sup: pose a second unit, of the same kind, to be divided into the same number of equal parts, those parts may also be expressed in the same fraction with the parts of the first unit. Thus, is read three halves. seven fourths. sixteen fifths. The denominator of the first fraction, shows that a unit has been divided into 2 equal parts, and the numerator expresses that three such parts are taken. Now, two of the parts make up one unit, and the remaining part comes from the 2d unit: hence the value of the fraction is lf; that is, one and one half. The denominator of the second fraction shows that a unit has been divided into four equal parts, and the numerator expresses that 7 such parts are taken. Four of the 7 parts come from one unit, and the remaining 3 from a second unit: the value of the fraction is therefore equal to 1}; that is, to one and three-fourths. In the third fraction, the unit has been divided into 5 equal parts, and 16 such parts are taken. Now, since each unit has been divided to 5 parts, 15 of the 16 parts make 3 units, and the nnining part is 1 part of a fourth unit. There silue of the fraction is 34: that is, three and one fifth. From what has been said, we con. clude : $59. 1st. That a fraction is the expression of one or more parts of unity. 2d. That the denominator of a fraction shows into how many equal parts the unit or single thing has been divided, and the numerator expresses how many of such parts are taken in the fraction. 3d. That the value of every fraction is equal to the quotient arising from dividing the numerator by the denominator. 4th. When the numerator is less than the denominator, the value of the fraction is less than 1. 5th. When the numerator is equal to the denominator, the value of the fraction is equal to 1. 6th. When the numerator is greater than the denominator, the value of the fraction is greater than 1. 1 1 QUESTIONS. $ 56. What does the unit 1 represent? If we divide it into two equal parts, what is each part called ? If it be divided into three equal parts, what is each part ? Into 4, 5, 6, &c. parts? What are such expressions called ? $ 57. Of how many parts is each fraction made up? What is the one above the line called ? the one below the line ? What does the denominator show? What does the numera. .tor show? $ 58. When the numerator and denominator are equal, what is the value of the fraction ? $59. Repeat the six principles. OF FEDFRAL MONEY. The coins of the United States are of gold, si and copper, and are of the following denomi 1. Gold-Eagle, half-eagle, quarter-eagle. 2. Silver-Dollar, half-dollar, quarter-dollar, dime, half-dime. 3. Copper-Cent, half-cent. If a given quantity of gold or silver be divided into 24 equal parts, each part is called a carat. If any number of carats, be mixed with so many equal carats of a less valuable metal, that there be 24 ca. rats in the mixture, then the compound is said to be as many carats fine as it contains carats of the more precious metal, and to contain as much alloy as it contains carats of the baser. For example, if 20 carats of gold be mixed with 4 of silver, the mixture is called gold of 20 carats fine, and 4 parts alloy. The standard for the gold coin is 22 carats of gold, 1 of silver and 1 of copper. The standard for silver coins is 1485 parts of pure silver to 179 of pure copper. The copper coins are of pure copper. TABLE OF FEDERAL MONEY. 10 Mills marked (m) make 1 Cent, marked ct. 10 Cents make 1 dime, d. 10 Dimes make 1 dollar, 10 Dollars make 1 eagle, E. 961. In this table 10 units of either denomination make one unit of the next higher denomination, and this is the same way that simple numbers increase from the right to the left, Ø 10. Therefore, the de. nominations of federal money here expressed may be added, subtracted, multiplied and divided, by the same rules that have already been given for simple numbers. |