92. STEP II.-To multiply by using the parts of the multiplier.

1. The multiplier may be made into any desired parts, and the multiplicand taken separately the number of times expressed by each part. The sum of the products thus found is the required product.

Thus, to find 9 times 12 we may take 4 times 12 which are 48, then 5 times 12 which are 60. 4 times 12 plus 5 times 12 are 9 times 12; hence, 48 plus 60, or 108, are 9 times 12. 2. When we multiply by one of the equal parts of the multiplier, we find one of the equal parts of the required product. Hence, by multiplying the part thus found by the number of such parts, we find the required product.

For example, to find 12 times 64 we may proceed thus:

(1.) Observe, that 124 +4 + 4; hence, 4 is one of the 3 equal parts of 12.

(2.) That 64 is taken 12 times by taking it 4 times +4 times + 4 times, as shown in the analysis.

(3.) That 4 times 64, or 256, is one of the 3 equal parts of 12 times 64. Hence, multiplying 256 by 3 gives 12 times 64, or 768.

3. In multiplying by 20, 30, and so on up to 90, we invariably multiply by 10 one of the equal parts of these numbers, and then by the number of such parts.

For example, to multiply 43 by 30, we take 10 times 43, or 430, and multiply this product by 3; 430 × 3 = 1290, which is 30 times 43.

We multiply in the same manner by 200, 300, etc., 2000, 3000, etc.; multiplying first by 100, 1000, etc., then the product thus found by the number of 100's, 1000's, etc.

93. PROB. II. To multiply by a number containing only one order of units.

EXPLANATION.-500 is equal to 5 times 100; hence, by taking 347, as in first step, 100 times, 5 times this result, or 5 times 34700, as shown in second step, will make 500 times 347. Hence 173500 is 500 times 347.

2. In practice we multiply first by the significant figure, and annex to the product as many ciphers as there are ciphers in the multiplier, as shown in (2); hence the following

94. RULE.-Multiply by the significant figure and annex as many ciphers to the result as there are ciphers in the multiplier.

96. PROB. III.—To multiply by a number containing two or more orders of units.

EXPLANATION.-1. The multiplier, 374, is analyzed into the parts 4, 70, and 300, according to (92).

2. The multiplicand, 539, is taken first 4 times = 2156 (86); then 70 times = 37730 (93); then 300 times = : 161700 (93).

3. 4 times + 70 times + 300 times are equal to 374 times; hence the sum of the partial products, 2156, 37730, and 161700, is equal to 374 times 539 = 201586.

4. Observe, that in practice we arrange the partial products as shown in (2), omitting the ciphers at the right, and placing the first significant figure of each product under the order to which it belongs. Hence the following

97. RULE.-I. Write the multiplier under the multiplicand, so that units of the same order stand in the same column.

II. Multiply the multiplicand by each significant figure in the multiplier, successively, beginning at the right, and place the right-hand figure of each partial product under the order of the multiplier used. Add the partial products, which will give the product required.

PROOF.-I. Repeat the work. II. Use the multiplicand as multiplier; if the results are the same the work is probably correct.

EXAMPLES FOR PRACTICE.

98. Copy examples from Arithmetical Table No. 1, page 16.

Multiplicand five figures, Multiplier three.

1. Take the multiplicands in order, commencing opposite 1, from columns A, B, C, D, E; B, C, D, E, F; C, D, E, F, G; D, E, F, G, H; and E, F, G, H, I.

2. Take the multipliers in each set from the three righthand columns used for multiplicands, the number immediately under the multiplicand.

Multiplicand six figures, Multiplier five.

1. Take the multiplicands in order from columns A, B, C, D, E, F; B, C, D, E, F, G; C, D. E, F, G, H; and D, E, F, G, H, I. 2. Take the multipliers in each set from the five right-hand columns used for multiplicand.

99. 1. A man left $4500 to his wife, $3254 to each of his five daughters, and the remainder of his property, amounting to $3860 to his only son; what was the value of his estate?

2. If you should buy 2682 barrels of flour, at $9 a barrel, and pay $15838 down, how much would you still owe for the flour? Ans. $8300.

3. I bought 8 barrels of sugar, at $54 a barrel; 3 barrels of it were spoiled by exposure, but the rest was sold at $72 a barrel; how much did I lose on the sugar? Ans. $72.

4. Sold 5 oxen at $75 each, 3 horses at $256 each, a carriage at $325, and a plow for $25; how much did I receive for the whole?

Ans. $1493.

5. There are 63 gallons in a hogshead; how many gallons in 8290 hogsheads? Ans. 522270. 6. If an acre yields 114 bushels of potatoes, how many bushels may be raised on 124 acres? Ans. 14136 bushels.

7. If 276 men can do a piece of work in 517 days, in what time could one man do the same work? Ans. 142692 days.

8. A man owns 2 orchards, in each of which there are 21 rows of trees, with 213 trees in each row; how many trees do both orchards contain ? Ans. 8946 trees. 9. If 2 tons of hay, worth $13 a ton, winter one cow, what will be the cost of wintering 348 cows? Ans. $9048.

10. I bought 14 cows at 39 dollars each, and 29 oxen at 63 dollars each; how much did I pay for all? Ans. $2373. 11. France contains 203736 square miles, and the population averages 176 per square mile; what is the entire population? Ans. 35857536.

12. A square mile contains 640 acres; find the cost of 36 square miles at $45 an acre. Ans. $1036800. 13. What is the cost of 5 yards of cloth at $2.25 a yard. SOLUTION. Since 1 yard costs $2.25, 5 yards will cost 5 times $2.25, which is $11.25.

Observe, that when the multiplicand contains cents, we multiply without regard to the period, and insert a period between the second and third figures of the result. The two figures at the right express the cents in the answer.

14. A farmer sold 57 bushels of beans at $2.36 per bushel, and 285 bushels of wheat at $1.75. How much did he receive for both? Ans. $633.27.

15. A fruit merchant bought 295 baskets of peaches at $1.25 a basket; finding that 43 baskets were worthless, he sold the rest at $1.75; how much did he make on the transaction?

16. A drover bought 94 head of cattle at $39 a head and 236 sheep at $3.89 a head. He sold the cattle at a gain of $9 a head and the sheep at a loss of $.75 a head; what was the total amount of the sale, and the gain on the transaction?