and divisor, is termed Cancellation. It depends on the principle explained in Arts. 58 and 59. NOTE. To cancel is to suppress or erase. When the same factor is omitted in both dividend and divisor, it is said to be canceled. RULE FOR CANCELLATION. When there are common factors in a dividend and its divisor, shorten the operation by canceling all the factors common to both : proceed with the remaining factors as the question may require. REM.-1. Canceling is merely dividing both dividend and divisor by the same number, which (Art. 59) does not alter the quotient. 2. The pupil should observe that one factor in the dividend will cancel only one equal factor in the divisor. 3. Some prefer to place the numbers forming the dividend on the right of a vertical line, and those forming the divisor on the left, 1. Multiply 42, 25, and 18, together, and divide the product by 21× 15. Ans. 60. 2. I sold 23 sheep at $10 each, and was paid in hogs at $5 each how many did I receive? Ans. 46. 3. How many yards of flannel at 35 cents a yard, will pay for 15 yards of calico at 14 cts.? Ans. 6 yards. 4. What is the quotient of 21×11×6×26, divided by 13×3×14×2? Ans. 33. 5. The factors of a dividend are 21, 15, 33, 8, 14, and 17; the divisors, 20, 34, 22, and 27: required the quotient. Ans. 49. 6. I bought 21 kegs of nails of 95 pounds each, at 6 cents a pound; paid for them with pieces of muslin of 35 yards each, at 9 cents a yard: how many pieces of muslin did I give? Ans, 38. NOTE. Other applications of Cancellation will be found in Fractions, Proportion, &c. The pupil will apply it more readily, when acquainted with Factoring. REVIEW.-619 REM. How are numbers arranged for cancellation? 616. What is cancellation? Upon what principle does it depend? What does cancel mean? When is a factor canceled? What is the rule? REM, What is canceling? VII. COMPOUND NUMBERS. TO TEACHERS.-While placing Fractions immediately after Simple Whole Numbers is philosophical, and appropriate in a Higher Arithmetic for Advanced pupils, the experience of the author convinces him that, in a hook for Young learners, Compound Numbers should be introduced here, nstead of after Fractions, as is done by some authors. His reasons are, 1st. The operations of Addition, Subtraction, Multiplication, and Division of Compound Numbers, are analogous to the same operations in Simr1 Numbers, and serve to illustrate the principles of the fundamental rulea The principle of Notation is the same in each. 2d. The subject of Fractions is important and difficult. Before studying it, most pupils require more mental discipline than is demanded in the ele mentary rules. This is acquired by the study of Compound Numbers. 3d. The general principles involved in their study, do not require a knowledge of fractions. The Examples involving fractions are few, and are introduced, (as they should be,) with other exercises in that subject. TEACHERS who prefer it, can direct their pupils to defer Compound Numbers until they have studied Fractions to page 169. DEFINITIONS. ART. 62. When two numbers have the same unit, they are of the same kind or denomination: thus, 3 dollars, 5 dollars, are of the same denomination; both dollars. When they have different units, they are of different denominations: thus, 3 dollars, and 5 cents, are of different denominations; dollars and cents. ART. 63. A simple number denotes things of the same unit value: thus, 3 yards, 2 dollars, 5 pints, are each simple numbers. All abstract numbers are simple. ART. 64. A compound number is two or more numbers of different unit values used to express one quantity: thus, 3 dollars 5 cents, 2 feet 3 inches, are each compound. REM.-1. In Compound Numbers, denomination or order, denotes the REVIEW.-62. When are two numbers of the same denomination? Give an example. When of different denominations? Give an example. 63. What does a Simple Number denote? Give an example. 64. What is a Compound Number? Give an example. name of the unit considered. Thus, dollar and cent are denominations of money; foot and inch, of length; pound and ounce, of weight. 2. Compound Numbers are analogous to Simple Numbers in this particular; a certain number of units of each order is collected into a group, and forms a unit of a higher order or denomination. But, They differ in this: that in compound numbers 10 units of one order do not uniformly make one of the next higher. 3. The simplest class of Compound Numbers is Federal money, because we pass from one denomination to another according to the scale of tens. FEDERAL OR UNITED STATES MONEY, ART. 65, Is the currency of the United States, established by the Federal Congress, in 1786. While U. S. money may be treated decimally, it is a species of Compound Numbers, being so regarded in ordinary business transactions. Its denominations, or the names of its different orders, are mill, cent, dime, dollar, eagle. Ten units of each denomination make one unit of the next higher denomination. The coins of the United States are of copper, silver, and gold. Their denominations are, 1st. Copper: cent, half cent. (3 cent piece, silver and copper.) 2d. Silver: dollar, half dollar, quarter dollar, dime, half dime 3d. Gold: $20 piece, eagle, half eagle, quarter eagle, three dollar piece, dollar. The mill is not coined. REVIEW.-64. REM. 1. What does denomination or order denote? 2. In what are Simple and Compound Numbers analogous? In what do they differ? 3. What is the simplest class of Compound Numbers ? NOTATION AND NUMERATION. ART. 66. Accounts are kept in dollars, cents, and mills; or in dollars, cents, and parts of a cent. Eagles and dollars are called dollars; dimes and cents, cents. Hundreds of dollars. Tens of dollars, or cagles. Tens of cents, or dimes. Cents. co Mills. NUMERATION TABLE. .0 4 3 .2 1 4 3.0 4 5 7 6.2 5 0 6 8 1.3 4 5 read 4 cents and 3 mills, or 43 mills. read 681 dollars 34 cents and 5 mills. The second line may also be read, 2 dimes, 1 cent, 4 mills; the fourth line, 7 eagles, 6 dollars, 5 cents. This method of reading is not customary. The third line may also be read, 304 cents and 5 mills, or 3045 mills; the fourth line, 7625 cents, or 76250 mills; the lower line, 68134 cents and 5 mills, or 681345 mills. A period (.), is used as a separating point, to separate the cents and dollars. Some use the comma. Thus, 2 dollars, 2 dimes, 2 cents, or 2 dollars and 22 cents, are written, $2.22 ART. 67. The Table shows that cents occupy the first two places to the right of dollars, and mills the place to the right of cents, the third from dollars. Hence the Rule for Numeration.—Read the number to the left of the period as dollars, and the first two figures on the right of the period as cents; and if there be a third figure, as mills. REVIEW.-65. What are the denominations of United States money? How many units of either denomination make a unit of the next highor? Repeat the table. How many cents in a half dime? In a quarter dollar? In a half dollar? In a dollar? Of what are the coins of the United States? Which are copper? Which silver? Which gold? ART. 68. RULE FOR NOTATION. Write the dollars as in whole numbers; place a period on the right of dollars, next to this write the cents, then the mills. If the cents are less than ten, place a cipher next to the dollars; if there are no cents, put two ciphers in the place of cents. EXAMPLES TO BE WRITTEN. 1. Twelve dollars, seventeen cents, eight mills. $12.17 8 2. Six dollars, six cents, six mills. 3. Seven dollars, seven mills. $ 6.06 6 4. Forty dollars, fifty-three cents, four mills. . $40.53 4 5. Two dollars, three cents. 6. Twenty dollars, two cents, two mills. 7. One hundred dollars, ten cents. 8. Two hundred dollars, two cents. 2.03 $100.10 $200.02 9. Four hundred dollars, one cent, eight mills. $400.01 8 REDUCTION OF U. S. MONEY. ART. 69. Reduction consists in changing the denominations or orders, without altering the value. 1st. Reduction Descending is changing numbers from a higher to a lower denomination; as, from dollars to cents. 2d. Reduction Ascending is changing numbers from a lower to a higher denomination; as, from cents to dollars. REVIEW.-66. In what denominations are accounts kept? What are eagles and dollars together called? What dimes and cents? How are dollars and cents separated? 67. What places do cents occupy? What place mills? What is the Rule for Numeration? 68. How write any sum of U. S. money, Rule? 69. In what does Reduction consist? What is Reduction descending? What ascending? 1 |