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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 ninoteen-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 ? What is a denominate number? What is the unit of 6 dollars? When two numbers have the same unit, what do you say of them? When they have different units ? Are 6 dollars and 4 dollars of the same denomination ? Are
4 dollars and 4 cents? What is the unit of each? Are several num bers of different denominations ofien connecied 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 of
pure silver, to 179 of pure copper.
The copper coins are of pure copper.
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.
1 Dime, d.
1 Dollar, 10 Dollars
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 ? În 7 dollars ?
NUMERATION TABLE FOR FEDERAL MONEY.
Thousands of dollars.
7, is read, 5 cents and 7 mills, or 57 mills. 1 6 4,
16 cents and 4 mills, or 164 mills. 6 2,1 20, 62 dollars 12 cents and no mills. 1 2 7,6 23, 127 dollars, 62 cents and 3 mills. 8 9 4 0,0 41, 8940 dollars, 4 cents and 1 mill.
As dimes are tens of cents the second line may either be read 16 cents and 4 mills, or 1 dime 6 cents and 4 mills. And as the eagles are tens of dollars the third line may be read 62 dollars and 12 cents, or 6 eagles 2 dollars and 12 cents.
Federal Money is generally read in dollars cents and mills.
Q. In numerating Federal Money, what is the figure on the right called? The second ? The third ?' The fourih? How is Federal Money generally read ?
REDUCTION OF FEDERAL MONEY. $ 47. REDUCTION of Federal Money consists in changing its denominations without altering its value. divided into two parts.
1st. To reduce from a higher denomination to a lower, as from dollars to cents.
2d. To reduce from a lower denomination to a higher, as from mills to dollars.
Q. What is reduction? How many kinds of reduction are there? Name them.
1. Reduce 25 Eagles 8 dollars 65 dimes and 35 cents, to the denomination of cents.
25 Eagles the highest denomination.
10 dollars make one eagle. 250 Product in dollars. add 8 the number in the denomination of dollars. 258
10 the number of dimes in a dollar. 2580 Product in dimes. add 65 the number in the denomination of dimes. 2645
10 number of cents in a dime. 26450 Product in cents.
35 cents to be added. 26485 Number of cents in 25 eagles 8 dollars 65 dimes
and 35 cents.
2. In 3 dollars 60 cents and 5 mills : how many mills? 3 dollars=300 cents,
60 cents to be added, 360=3600 mills, to which add the 5 mills.
Ans. 3605. 3. In 37 dollars 37 cents 8 mills : how many mills ?
Ans. 37378. 4. In 375 dollars 99 cents 9 mills : how many mills ?
Ans. 375999. 5. How many mills in 67 cents ?
Ans. 6. How many mills in $54 ?
Ans. 54000. 7. How many cents in $125 ?
Ans. 12500. 8. In $400, how many cents ? How many mills ? 9. In $375, how many cents ? How many mills ? 10. How many mills in $4? In $6? In $10,14 cents ? 11. How many mills in $40,36 cents 8 mills ?
§ 48. As we change dollars into cents by adding two ciphers, and cents into mills by adding one, it follows that, to change mills into dollars cents and mills, we have the following
RULE. Cut of the right hand figure for mills, and the figures to the left will be cents. Then cut off the two next figures for cents, and the remaining figures to the left will be dollars.
The reason of the rule is this: by cutting off the first right hand figure, we in fact, divide by 10, and thus reduce the mills to cents. Then by cutting off the next two figures, we divide by 100; and thus reduce the cents to dollars.
The comma, or separatrix, is generally used to separate the cents from the dollars. It is not usual to place the comma between the cents and mills. Thus, $67,25 6 is read 67 dollars 25 cents and 6 mills.
Q. How do you change mills into cents? How do you change cents into dollars? How do you separate the mills from the cents? How the cents from the dollars ?
1. How many dollars cents and mills, are there in 67897 mills?