The Product, or number produced by multiplying the other two terms together.

I. When the multiplier is 12 or less.

RULE 1.-Set the multiplier under the right-hand figure of the multiplicand.

2.-Multiply each figure separately from the right to the left, setting down the right-hand figure and carrying the rest, as in addition.

Multiplication is a short method of addition, since, if we should place six nines under each other, and add them together, the result would be the same as in multiplying 9 by 6.

3.-Set down the whole of the last product.

PROOF.-Add together the figures in each term, observing to cast out the nines as they arise in summing; multiply the remainders from the multiplier and multiplicand together, casting out the nines again; if the excess thence arising equal the excess of nines from the product, the work is supposed to be

correct.

Sometimes it will do to reverse the factors, i. e. make the multiplicand the multiplier, and the multiplier the multiplicand; or to take a number one less than the real multiplier, and add the multiplicand to the product.

How do you work when the multiplier is less than 12-1st? 2d? 3d? How

do you prove multiplication?

26. Multiply 486062794513 by all the numbers from 1 to 12, and prove your work.

27. How many peaches in 9 baskets, allowing each basket Ans. 3582 peaches.

to contain 398?

28. How far will a ship sail in 12 days, at the rate of 249 miles per day? Ans. 2988 mi. 29. How many days in 8 years, allowing 365 days for each year, and 2 extra days for leap years? Ans. 2922 days. 30. Sold 3409 tons of coal at $5 per ton; what amount of money is due me? Ans. $17045.

II. When the multiplier is more than 12, and consists of two or more significant figures.

RULE 1.-Place the multiplier under the multiplicand on the right, units under units, tens under tens, etc.

2.-Multiply the multiplicand by each figure of the multiplier separately, beginning each time exactly under the figure you use; set down and carry as before.

How do you work when the multiplier is more than 12, and consists of two or more significant figures-1st? 2d? *3d?

III. When the multiplier is a composite number. (See Definitions, page 10.)

RULE.-Multiply by one of the factors first, and then multiply the product thence arising by the other factor. PROOF.-AS before, or reverse the factors, or take other factors.

51. Multiply 9486 by 36.
52. Multiply 3894064 by 42.
53. Multiply 9408623 by 48.
54. Multiply 8940862 by 49.

Ans. 341496.

Ans. 163550688.

Ans. 451613904.

Ans. 438102238.

55. Multiply 3948672 by 56; by 64; by 66; by 72; by 77; by 81; by 84; by 88; by 96; by 108; by 121; by 132; by 144; and prove your work.

56. Bought 346 pieces of muslin, 24 yards in a piece; how many yards? Ans. 8304 yds. 57. In a certain city there are 3482 blocks of houses, and 48 houses in a block; how many houses?

Ans. 167136 houses.

IV. When there are ciphers on the right of either or both of the factors.

RULE.-Neglect the ciphers on the right of the factors, and work with the rest of the figures, as before; and to the right of the product annex all the ciphers neglected. PROOF-As before.

If the multiplier is 10, 100, 1000, etc., you have only to add its ciphers to the multiplicand, and you have the product. 63. Multiply 4862 by 10.

Ans. 48620.

64. Multiply 9640 by 100; by 1000; by 10000.

Ans. 964000; 9640000; 96400000.

What is the rule for multiplying by a composite number? What is the r where there are ciphers on the right of either or both factors? How do you prove the work in both these cases?

V. To multiply integers and decimals.

RULE. Place the multiplier under the right-hand figures of the multiplicand, and multiply as before; point off on the right of the product as many figures for decimals as there are decimal places in both factors; if you have not enough, prefix ciphers to the product.

70. Multiply 3.4689 by 37.48; by 4.08; by 9.006; by .007; by .0046; by .0009; by 49.047; by 67.4; by 9.64; and prove the work.

VI. To multiply dollars, cents, and mills.

RULE. Place the factors and multiply as before; in the product place the point between the different denomi

nations.

PROOF.-As before.

EXAMPLES.

71. Multiply $24.22 by 2.

$ cts. 24.22 2

Ans. $48.44cts.

72. Multiply $32.26.4 by 3.

Ans. $96.79.2.

Ans. $89.46. Ans. $183.12.5.

73. Multiply $22 36cts. and 5m. by 4. 74. Multiply $36 62cts. and 5m. by 5. 75. Multiply $126 87cts. and 4m. by 6; by 7; by 8; by 9; by 10; by 11; by 12; by 25; by 34; by 136; by 245; and prove your work.

76. How much for 12 yards of cloth at $2 62cts. and 5m. per yard? Ans. $31.50.