5. Multiply 17 by 18, and divide the product by 6.

(6X3=18), then cross the factor 6, which is common to both mul

tiplier and divisor: after which, multiply 17 by 3.

6. In 15 times 8, how many times 4? 7. In 24 times 4, how many times 8? 8. In 37 times 15, how many times 5? 9. Multiply 36 by 40, and divide the multiplied by 8.

OPERATION.

Ans. 30.

Ans. 12.

Ans. 111.

product by 30

Dividend, 36×40_6×3×2×5×4×2

=

Divisor, 30 X 8 6×3×4×2

= = 6.

SOLUTION.-Indicate the operation to be performed, then resolve (separate) the numbers into factors. Now the quotient will not be changed (Art. 59) by dividing both divisor and dividend by the same number, which is done by erasing the same factors in both..

mon factor 6, divide each by 6, and write the quotients 6 and 5 as in the operation cancel 36 and 30. Next, canceling the common factor 5 in both dividend and divisor, the result is 6, as before.

10. In 36 times 5, how many times 15?

11. In 14 times 9, how many times 6?

Ans. 12.

Ans. 21.

ART. 616. The process of shortening the operations of arithmetic, by omitting equal factors from the dividend

REVIEW.-60. If a number be multiplied, and the product divided by the same number, what will be the quotient? On what principle? If a number be divided and the quotient multiplied by the same number, what will the product be? On what principle?

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 product by 21x15.

divide the 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.-61. 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 book for Young learners, Compound Numbers should be introduced here, instead 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 Simple Numbers, and serve to illustrate the principles of the fundamental rules. 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 elementary 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.

10 mills,

10 cents

10 dimes

TABLE.

10 dollars

Also, 5 cents

1 eagle,

E.

make one-half dime.

25 cents

50 cents

75 cents

100 cents

⚫ one-quarter of a dollar.

one-half of a dollar. three-quarters of a dollar. one dollar.

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 eagles.
Dollars.

Tens of cents, or dimes.

Cents.
∞ Mills.

NUMERATION TABLE.

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 21 cents and 4 mills, or 214 mills.
read 3 dollars 4 cents and 5 mills.
read 76 dollars and 25 cents.

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 higher ? 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?