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.

Q. When the numerator and denominator are equal, what is the value of the fraction? What is the value of the fraction three halves? Of seven fourths? Of sixteen fifths? Of eighteen sixths? Of twenty-five sevenths? Repeat the six principles. Write the fraction nineteen-fortieths:-also, 60 fourteenths-18 fiftieths-16 twentieths17 thirtieths-41 one thousandths-69 ten thousandths-85 millionths -106 fifths.

OF DENOMINATE NUMBERS.

§ 45. Simple numbers express a collection of units of the same kind, without expressing the particular value of the unit. For example, 40 and 55 are simple numbers, and the unit is 1, but it is not expressed whether the unit is 1 apple, 1 pound, or 1 horse.

A DENOMINATE number expresses the kind of unit which is considered. For example, 6 dollars is a denominate number, the unit 1 dollar being denominated, or named.

When two numbers have the same unit, they are said to be of the same denomination: and when two numbers have different units, they are said to be of different denominations.

For example, 10 dollars and 12 dollars are of the same denomination; but, 8 dollars and 20 cents, express numbers of different denominations, the unit of 8 dollars being 1 dollar, and of 20 cents, 1 cent.

Several numbers of different denominations are often connected together, forming a whole, as 3 dollars 15 cents.

Q. What do simple numbers express? number? What is the unit of 6 dollars? the same unit, what do you say of them? units? Are 6 dollars and 4 dollars of the

What is a denominate When two numbers have When they have different same denomination? Are

4 dollars and 4 cents? What is the unit of each? Are several num bers of different denominations often connected together? Give an example.

OF FEDERAL MONEY.

§ 46. Federal money is the currency of the United States. Its denominations, or names, are Eagles, Dollars, Dimes, Cents, and Mills.

The coins of the United States are of gold, silver, and copper, and are of the following denominations.

1. Gold-Eagle, half-eagle, quarter-eagle.

2. Silver-Dollar, half-dollar, quarter-dollar, dime, halfdime.

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 carats 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 of the gold coin in the United States, is 22 carats of gold, 1 of silver, and 1 of copper. The standard for silver coins is 1489 parts

to 179 of pure copper.

The copper coins are of pure copper.

of

pure silver,

Q. What is the currency of the United States? What are its denominations? What are the coins of the United States? Which gold? Which silver? Which copper? What do you understand by gold 20 carats fine? What is the standard of the gold coin? What of the silver coin? What of the copper?

TABLE OF FEDERAL MONEY.

10 Mills marked (m) make 1 Cent, marked ct.
10 Cents

d.

1 Dime, -
1 Dollar,

1 Eagle,

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. Therefore,

The denominations of federal money here expressed may be added, subtracted, multiplied, and divided, by the same rules that have already been given for simple numbers. From the table it appears,

1st, That cents may be changed into mills by annexing a cipher.

Thus, 8 cents are equal to 80 mills.

2d. That dollars may be changed into cents by annexing two ciphers, and into mills by annexing three.

For example, 12 dollars are equal to 1200 cents, or to 12000 mills. The reason of these rules is evident, since 10 mills make a cent, 100 cents a dollar, and 1000 mills a dollar.

Q. Repeat the table. How many units of either denomination make one of the next higher? How do simple numbers increase from the right to the left? How may Federal Money be added, subtracted, multiplied, and divided? How may cents be changed into mills? How may dollars be changed into cents? How into mills? To how many cents are 12 dollars equal? To how many mills are they equal? How many cents in 4 dollars? How many in 6 dollars? How many mills in 9 dollars? How many mills in 5 dollars? How many cents in 3 dollars? In 8 dollars? In 7 dollars?

57, is read, 5 cents and 7 mills, or 57 mills.

164,

62, 1 2 0,

127, 62 3,

8940,

8 9 4 0, 0 4 1,

DEMONSTRATION OF THE RULE.

The number 700=100×7. Hence it is a composite number of which the factors are 100 and 7.

In striking off the two figures 89, from the right of the dividend, we divide it by 100; we then divide the 673 by the other factor 7. We then multiply the remainder 1 by 100 and add 89 to the product, giving 189 for the true remainder, (see § 37.)

2. Divide 8749632 by 37000.

371000)87491632(236

74

Q. How do you divide by 10, 100, 1000, &c.? (see § 38.) Which part is the quotient? Which part is the remainder? When there are ciphers on the right of the divisor, how do you form the true remainder?

APPLICATIONS IN DIVISION.

OPERATION.

1. Divide 80 dollars equally among four men. Here the 80 dollars is to be divided into 4 equal parts, and the quotient 20 dollars expresses the value of one of the equal parts.

4)80

20 dollars.

2. Four persons buy a lottery ticket; it draws a prize of 10000 dollars: what is each one's share?

3. A person dying leaves an estate of 4500 dollars to be divided equally among 5 children: what is each one's share? Ans. 900 dollars.

4. There are 1560 eggs to be packed in 24 baskets: how many eggs will be put in each basket?

Ans.

5. What number must be multiplied by 124 to produce 40796 ? Ans. 329.

6. How many times can 24 be subtracted from 1416?

Ans. 7. The sum of 19125 dollars is to be distributed among a certain number of men, each is to receive 425 dollars: how many men are to receive the money? Ans.

8. By the census of 1840 the whole population of the 26 States was 16,890,320: if each one had contained an equal number of inhabitants, how many would there have been in each state? Ans. 649,62718.

9. If a man walks 12775 miles in a year, or 365 days, how far does he walk each day? Ans. miles. 10. A farmer sells a drove of sheep for 2 dollars a head, and receives 1250 dollars: how many sheep did he sell? Ans. 625.

11. It is computed that the distance to the sun is 95,000,000 of miles, and that light is 8 minutes travelling from the sun to the earth: how many miles does it travel per minute?

Ans.

12. By the census of 1840 it appeared that the City of New York contained 312710 inhabitants; allowing 5 to each house, how many houses were there in the city at that time? Ans. 62,542.

13. A merchant has 5100 pounds of tea, and wishes to pack it in 60 chests: how many pounds must he put in each chest?

Ans.

14. A person goes to a store and buys a piece of cloth containing 36 yards, for which he pays 288 dollars: how much does he pay per yard? Ans. dollars. 15. There are 7 days in a week: how many weeks in Ans. 52 weeks and 1 day over. 16. There are 24 hours in a day: how many days in 2040 hours? Ans. days. 17. Twenty-three persons dined together, their bill was 92 dollars. how much had each one to pay ?

a year of 365 ?

Ans. 4 dollars.