GENERAL REMARKS. § 40. Numeration, Addition, Subtraction, Multiplication, and Division, are called the five ground rules of Arithmetic. Q. How many principal rules are their in Arithmetic? What are they? Can Multiplication be performed by Addition? Can Division De performed by Subtraction? By how many rules, then, may all the operations in Arithmetic be performed? § 41. The preceding rules furnish answers to the following questions. Ques. 1. When the cost of each one of several things is given, how do you find their entire cost? Ans. Add the costs of the several things together, the sum will be the entire cost. What is the entire cost of a bag of coffee at 6 dollars, a chest of tea at 4 dollars, a box of raisins at 2 dollars, and a barrel of sugar at 12 dollars? Ans. 24 dollars. Q. 2. When you have two unequal numbers, how do you find their difference? A. By subtracting the less from the greater. Q. 3. When the subtrahend and remainder are given, or known, how do you find the minuend? A. By adding the remainder and subtrahend together. Hence the following principles. 1st. If the sum of two numbers be diminished by one of them, the remainder will be the other number. 2d. The less of two numbers added to their difference, will give the greater. The sum of two numbers is 56, one of the numbers is 12: what is the other? Ans. 44. The less of two numbers is 25, and their difference 30: what is the greater? Ans. The less of two numbers is 35, and their difference 35: what is the greater? Ans. 70. Q. 4. When you have the cost of a single thing, how will you find the entire cost of any number of things at the same rate? A. Multiply the cost of the single thing by the number of things. What is the cost of 35 pears at 2 cents each? What is the cost of 45 yards of cloth at 3 dollars per yard? Q. 5. When you know the number of things, and their entire cost, how do you find the cost of a single thing of the same kind? A. Divide the entire cost by the number of things, the quotient will be the cost of a single thing. If 60 oranges cost 360 cents, how much do they cost apiece? If 40 yards of cloth cost 200 dollars, how much is it a yard? APPLICATIONS IN THE PRECEDING RULES. 1. A Farmer sells a yoke of oxen for 90 dollars, 3 cows for 25 dollars each, 9 calves for 4 dollars each, and 65 sheep at 3 dollars a head. How much did he receive for them all? Ans. dollars. 2. The sum of two numbers is 365, one of the numbers is 221; what is the other number? Ans. 144. 3. The difference of two numbers is 95, the less number is 327; what is the greater number? Ans. 4. A farmer sells 4 tons of hay at 12 dollars per ton, 80 bushels of wheat at 1 dollar per bushel, and takes in part payment a horse worth 65 dollars, a wagon worth 40 dollars, and the rest in cash. How much money did he receive? Ans. 23 dollars. 5. A farmer has 14 calves worth 4 dollars each, 40 sheep worth 3 dollars each; he gives them all for a horse worth 150 dollars: does he make or lose by the bargain? Ans. He loses dollars. 6. The product of two numbers is 51679680, and one of the factors is 615: what is the other factor? Ans. 84032. Ans. 2080353420. Ans. 7. When the divisor is 67941, and the quotient 30620, what is the dividend? 8. When the dividend is 1213193, the quotient 37, what is the divisor? 9. A piece of cloth containing 65 yards costs 455 dollars: what does it cost per yard? Ans. dollars. 10. A man has 6 children, all of whom are married, and each has four children; two of these grand-children are § 44. 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 suppose 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, 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 1; 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 13; 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 into 5 parts, 15 of the 16 parts make 3 units, and the remaining part is 1 part of a fourth unit. Therefore, the value of the fraction is 31: that is, three and one fifth. The value of the fourth fraction is three, and of the fifth, three and four-sevenths. From what has been said, we conclude: 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 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. 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? 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 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 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. 10 Mills marked (m) make 1 Cent, marked ct. 10 Cents 1 Dime, 1 Dollar, |