GENERAL PRINCIPLES FROM THE FOREGOING RULES. 47. Numeration, Addition, Subtraction, Multiplication, and Division, are called the five ground rules of Arithmetic. 48. When the cost of each of several things is given, their entire cost is found by adding the costs of the several things together. Ex. 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. 2. What is the entire cost of six sheep at 12 dollars, a cow at twenty dollars, a horse at one hundred and fifty dollars, and a yoke of oxen at eighty dollars ? 3. What is the entire cost of 6 calves at 14 dollars, 25 sheep at 50 dollars, and one cow at 26 dollars ? 49. The difference between two numbers is found by subtracting the less from the greater : or, if they are equal, either may be subtracted from the other. Ex. What is the difference between 37 and 23 ? Also between 40 and 40 ? 50. If the subtrahend and remainder are known, the miuuend may be found by adding the remainder to the subtrahend. Hence, the following principles : 1st. If the sum of two numbers be diminished by one of them, the remainder will be the other number. 47. How many principal rules are thero in Arithmetic? Wha are they? Can multiplication be performed by addition? Can di vision be performed by subtraction ? By how many rules, then may all the operations in Arithmetic be performed? 48. When the cost of each of several things is given, how do yor find their entire cost? 49. How do you find the difference between two unequal num bers? Between two equal numbers ? 50. How do you find the minuend, when the subtrahend and re mainder are given ? If the sum of two numbers be diminished by one of them, what will the remainder be? If the less of two num bors be added to their difference, what will the sum be? 2d. The less of two numbers, added to their difference, will give the greater. Ex. 1. The sum of two numbers is 56, one of the numbers is 12: what is the other? Ans. 44. 2. The less of two numbers is 25, and their difference 30: what is the greater? Ans. 3. The less of two numbers is 35, and their difference 35 : what is the greater ? Ans. 70. 51. Knowing the price of a single thing, the cost of any number of things may be found by multiplying the number of things by the price of one of them. Ex. 1. What is the cost of 35 pears at 2 cents each ? What is the cost of 45 yards of cloth at 2 dollars a yard ? Of 65 yards of riband at 30 cents a yard ? 52. When the multiplier is 1 the product will be equal to the multiplicand. When the multiplier is greater than 1, the product will be as many times greater than' the multiplicand as the multiplier is greater than unity. Ex. 1. If 65 be multiplied by 1, what will the product be ? If it be multiplied by 2 what will the product be? How many times greater than 65 ? 2. If 17 be multiplied by 3, what will the product be ? How many times greater than 17 ? 53. When we know the number of things and their entire cost, the cost of a single, thing may be found by dividing the entire cost by the number of things. Ex. 1. If 6 oranges cost 36 cents, how much do they cost apiece? 2. If 4 yards of cloth cost 20 dollars, how much is it a yard? 51. Knowing the price of a single thing, how do you find the cost of several things of the same kind ? 52. When the multiplier is 1, how will the product compare with the multiplicand? When it is greater than i, how will they compare with each other? 53. When you know the number of things and their entire cost, how will you find the cost of a single thing? 3. If 6 yards of cloth cost 42 dollars, how much will it cost a yard ? 54. If the divisor is equal to the dividend the quotient will be 1, and the quotient will be as many times greater than 1, as the dividend is greater than the divisor. Ex. 1. If 15 be divided by 15, what will the quotient be? If 18 be divided by 18, what is the quotient ? 2. If 30 be divided by 6, what will the quot nt be? How many times is 30 greater than 6 ? 55. A prime number is one which cannot be exactly divided by any number except itself and unity. Thus, 1, 2, 3, 5, 7, 11, 13, 17, &c., are prime numbers. 56. The product of two or more prime numbers will be exactly divisible only by one or other of the fac. tors. 57. If an even number be added to itself any number of times, the sum will be even : hence, if one of the factors of a product be an even number, the product will be even. 58. An odd number is not divisible by an even number, nor is a less number exactly divisible by a greater. 59. Any number is divisible by 2, if the last signifi. cant figure is even; and is divisible by 4 if the last two significant figures are divisible by 4. 60. Any number whose last figure is 5 or 0, is exactly divisible by 5; and any number whose last significant figure is 0, is exactly divisible by 10. 54. If the divisor is equal to the dividend, what will the quatient be ? Generally, how will the quotient compare with 1 ? 55. What is a prime number? Give an example. 56. By what numbers only will the product of prime factors be divisible? 57. If an even number be multiplied by a whole number, will the product be odd or even ? 58. Is an odd number divisible by an even number? 59. Whut numbers are exactly divisible by 2? What numbers by 4 ? 60. If the last figure of a number be 5 or 0, by what number may it be divided ? OF DENOMINATE NUMBERS. 61. 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 they do not express 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. 62. 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. 63. When all the units of a number are of the same kind, it is called a simple denominate number. But several numbers of different denominations are often connected together, forming a single expression, as 3 dollars 15 cents. Such are called compound denominate numbers. 64. In Federal money we pass from one denomination to another according to the scale of tens, as in simple numbers; but in other denominate numbers, such, for example, as yards, feet, and inches, the scale is different, and we must consider how many units of each denomi. nation make one unit of the next higher. 61. What do simple numbers express ? What is a denominato number? What is the unit of 6 dollars? 62. 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 ? 63. What is a simple denominate number? What is a compound denominate number? Give an example. 64. Iu Federal money, what is the scale in passing from one devomination to another? How does this comparo with the scale in simple numbers? How is it in other denomipate numbers? What is always to be considered ? OF FEDERAL MONEY. [Before proceeding further, let the pupil study carefully from Art. 107 to Art. 109, page 107.) 65. Federal money is the currency of the United States. Its denominations, or names, are Eagles, Dol. lars, 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, 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 such a number of 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. 66. 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. The eagle contains 270 giuins of standard gold; the dollar 416 grains of standard silver; and the cent 11 pennyweights of copper. 65. What is the currency of the United States? What are its denominations ? What aro the coins of the United States? Which gold? Which silver? Which copper?: What do you understand by gold 20 carats fine? 66. What is the standard of the gold coin? What of the silver? What of the copper? What is the value, in gold, of the eagle? What is the value, in silver, of the dollar? What in copper of the cent? |