(1.) Therefore, 9,873 multiplied by 24 gives a product of 236,952.

(m.) RULE.-Write the multiplier under the multiplicand so that units may stand under units, tens under tens, &c.

Beginning with the unit figure, multiply each figure of the multiplicand by each figure of the multiplier, successively, / observing to write the first figure of each partial product directly under its multiplier, writing down and carrying as in addition.

Add the partial products, and their sum will be the product sought.

(n.) PROOF.-I. Interchange the multiplier with the multiplicand, and multiply as before; if the two products are the same, the work is supposed to be correct.

* II. Divide the product by the multiplier, and if the quotient equals the multiplicand, the work is correct.

FIRST METHOD OF PROOF.

MODEL OPERATION.

QUESTIONS.-What is Multiplication? (46.) What is the multiplicand? (47.) What is the multiplier? (48.) What is the product? (49.) Which terms are called factors? (50.) What is the sign of

*NOTE.-This method of proof presupposes a knowledge of division; but, as the pupil should already have a knowledge of the fundamental operations, and as it is the method usually adopted in business, we think it best to insert it here.

multiplication? (51.) Repeat the analysis of multiplication. (53.) Repeat the rule. (53., m.) Give both methods of proof. (53., n.) What is the difference between addition and subtraction? (31.,) (39.) What are the terms of subtraction? (38.,) (39.,) (40.,) (41.)

LESSON III.

PRACTICAL EXAMPLES CONTAINING BUT ONE ANALYTICAL STEP.

What will 1983 gallons of wine cost at 97 cts. a gallon? MODEL OPERATION.* 1983 No. of gallons.

97 cents.

13881

17847

192,351 cents.

ANALYSIS.-Arithmetical Formula.—If 1 gallon of wine cost

97 cents, 1,983 gallons will cost 1,983 times 97 cents, which are 192,351 cents.

Therefore, if 1 gallon of wine cost 97 cents, 1,983 gallons will cost 192,351 cents.

EXERCISE.

Require the pupil to compose and analyze ten problems similar to the model.

LESSON IV.

PRACTICAL EXAMPLES

COMBINING ADDITION, SUBTRACTION, AND MULTIPLICATION, AND WHICH CONTAIN TWO OR MORE

ANALYTICAL STEPS.

I purchased of a farmer 138 bushels of potatoes at 46 cents a bushel, and 37 bushels of corn at 96 cents a bushel;

NOTE.-Throughout the work the name of each denominate number is appended to it for the purpose of aiding the memory, the analysis in multiplication and division determining which number ought to be used as abstract.

I sold the farmer in return 63 yards of muslin at 13 cents per yard, and 41 yards of calico at 17 cents; how much money must I pay to balance the account?

(a.)

MODEL OPERATION.

138-No. bushels potatoes. 37=No. bushels corn.

46 cents.

828 552

(b.)

96 cents.

222

333

(c.)

6,348 cts., cost of potatoes. 3,552 cts., cost of corn,

6348 cents, cost of potatoes. 63-No. yds. muslin,

(d.)

13 cents.

9900 cents, cost of articles purchased.
1516 66
paid with articles in return.

7384 cents, balance due to farmer.

*ANALYSIS.

ANALYTICAL STEP. (a.) Find the cost of the potatoes.

*NOTE TO THE TEACHER.-The pupil should be required to give the full forms until he can do so promptly and accurately.

Elementary Question.—If I purchase 1 bushel of potatoes for 46 cents, what will 138 bushels cost?

Arithmetical Formula.-If 1 bushel of potatoes cost 46 cents, 138 bushels, &c.

ANALYTICAL STEP.-(b.) Find the cost of the corn.

Elementary Question.—If I purchase 1 bushel of corn, &c. Arithmetical Formula.—If 1 bushel of corn costs, &c.

ANALYTICAL STEP.-(c.) Find the cost of both the corn and the potatoes.

Elementary Question.-If the potatoes cost 6348 cents, and the corn 3552 cents, what will both cost?

Arithmetical Formula.-If the potatoes, &c.

ANALYTICAL STEP.-(d.) Find the amount received for the muslin.

Elementary Question.-If one yard of muslin costs, &c. Arithmetical Formula.—If one yard of muslin costs, &c. ANALYTICAL STEP.-(e.) Find the amount received for the

calico.

Elementary Question.—If one yard of calico costs, &c.

Arithmetical Formula.—If one yard of calico costs, &c. ANALYTICAL STEP.—(ƒ.) Find the amount received for both muslin and calico.

Elementary Question.-If the muslin costs, &c.

Arithmetical Formula.-If, &c.

ANALYTICAL STEP.-(g.) Find the balance due.

Elementary Question.-If, &c.

Arithmetical Formula.-If, &c.

CONCLUSION.-If I purchase of a farmer 138 bushels of potatoes, and &c.

EXERCISE.

Require the pupil to compose and analyze ten problems similar to the model.

SECTION VI.

DIVISION

LESSON I.

54. Division is the process of finding how many times one number is contained in another, or of separating. one number into as many parts as there are units in another. 55. The Dividend is the number to be divided. 56. The Divisor is the number by which the dividend is divided.

57. The Quotient is the number of times the divisor is contained in the dividend.

58. There are two methods of division, called Long Division and Short Division.

59. Short Division is the method generally used when the divisor does not exceed twelve.

60. In Long Division each step of the process is written, and it is the method generally used when the divisor is greater than twelve.

61. A Remainder in Division is that part of the dividend which remains undivided.

62. The SIGN of DIVISION is a short horizontal line, with a point above and another below it (÷), and when placed between numbers, it indicates that the number before it is to be divided by the number after it; thus, 20÷÷4=5, to be read, 20 divided by 4 is equal to 5.

(a.) Division is also expressed by writing the dividend above, and the divisor below, a short horizontal line; thus, 12=4, shows that 12 divided by 3 equals 4. This is sometimes called the fractional sign.