multiplicand, and add together its successive figures toward the right, till the sum obtained equals or exceeds the number nine. If it equals it, drop the nine, and begin to add again at this point, and proceed till you obtain a sum equal to, or greater than, nine. If it exceeds nine, drop the nine as before, and carry the excess to the next figure, and then continue the addition as before. Proceed in this way, till you have added all the figures in the multiplicand and rejected all the nines contained in it, and write the final excess at the right hand of the multiplicand.

Proceed in the same manner with the multiplier, and write the final excess under that of the multiplicand. Multiply these excesses together, and place the excess of nines in their product at the right.

Then proceed to find the excess of nines in the product obtained by the original operation; and, if the work is right, the excess thus found will be equal to the excess contained in the product of the above excesses of the multiplicand and multiplier.

NOTE.

This method of proof, though perhaps sufficiently sure for common purposes, is not always a test of the correctness of an operation. If two or more figures in the work should be transposed, or the value of one figure be just as much too great as another is too small, or if a nine be set down in the place of a cipher, or the contrary, the excess of nines will be the same, and still the work may not be correct. Such a balance of errors will not, however, be likely to occur.

17. What will 365 acres of land cost at 73 dollars per acre?

Ans. $26645.

18. What will 97 tons of iron cost at 57 dollars a ton?

Ans. $5529. 19. What will 397 yards of cloth cost at 7 dollars per yard? Ans. $2779. 20. What will 569 hogsheads of molasses cost at 37 dollars per hogshead? Ans. $21053. 21. If a man travel 37 miles in one day, how far will he travel in 365 days? Ans. 13505 miles. 22. If a vessel sails 169 miles in one day, how far will she sail in 144 days?

23. What will 698 barrels of flour cost at 7 dollars a barrel?

24. What will 376 lbs. of sugar cost at 13 cents a pound?

Ans. 4888 cts.

25. What will 97 lbs. of tea cost at 93 cents a pound?

Ans. 9021 cts.

26. If a regiment of soldiers consists of 1128 men, how many men are there in an army of 53 regiments?

Ans. 59784.

27. What is the product of 75432 × 47. Ans. 3545304. 28. What is the product of 76785316 × 7615.

Ans. 584720181340.

29. What is the product of 67853 × 8765.

Ans. 594731545.

30. What is the product of 3812345 × 31243.

Ans. 119109094835.

31. What is the product of 40670007 × 10002.

Ans. 406781410014.

32. What is the product of 31235678 × 10203.

Ans. 318697622634.

33. What is the product of 76786321 × 3007.

Ans. 230896467247.

34. What is the product of 6176777 × 22222.

Ans. 137260338494.

35. What is the product of 7060504 × 30204.

44. A full-grown elm will, it is computed, yearly, on an average, produce three hundred twenty-nine thousand three hundred and seventy-five seeds. How many seeds will three such trees produce in fifty-three years. Ans. 52370625.

45. John Alden can plant 3 plats of corn, containing 11 rows of 67 hills each, in 1 day, and Loring Blanchard can

plant twice as much in the same time. How many hills can Blanchard plant in a month of 26 working days? Ans. 114972.

46. If the multiplicand be three hundred and seventy-five millions two hundred and ninety-six thousand three hundred and twenty-one, and the multiplier seventy-nine thousand and twenty-four, what will be the product?

Ans. 29657416470704.

64. When the multiplier is a composite number.

Ex. 1. What cost 35 acres of land at 316 dollars an acre?

OPERATION.

316 dollars, cost of 1 acre.
7

2212 dollars, cost of 7 acres.
5

Ans. 11060 dollars.

The factors of 35 are 7 and 5. Now, if we multiply the price of one acre by 7, we get the cost of 7 acres; and, then, by multiplying the cost of 7 acres by the factor 5, it is evident, we obtain the Hence, when the multiplier is a

11060 dollars, cost of 35 acres.

cost of 5 times 7 acres, or 35 acres. composite number, we may

Multiply the multiplicand by one of the factors of the multiplier and the product thus obtained multiply by another, and so on until each of the factors has been used as a multiplier; and the last product will be the one sought.

EXAMPLES.

2. Multiply 3121 by 81, using its factors.

3. What will 63 horses cost at 175 dollars each?

4. A certain house contains 87 windows, and each window has 32 panes of glass. How many panes in the whole house? Ans. 2784.

5. What is the product of 47134987 by 56?

Ans. 2639559272.

6. If a garrison consume 6231 pounds of bread in 1 day, how many pounds will the same consume in 144 days?

Ans. 897264 pounds.

65. When there are ciphers on the right in the multiplier

or multiplicand, or both.

Ex. 1. In 1 yard there are 36 inches; how

OPERATION.

Multiplicand 36

Multiplier

10

Product

360 Or thus: 360,

36

100

3600

many inches in

Ans. 360, 3600.

We annex one cipher to the multiplicand to multiply it by 10, and two ciphers to multiply it by 100; since annexing one cipher removes each figure of the multiplicand one place to the left, and thus increases it 10 times; annexing two ciphers removes each figure two places to the left, and increases it 100 times; and so on, each additional cipher having the effect to increase its value 10 times (Art. 30).

3600.

2. What will 700 bales of cotton cost at 40 dollars per bale? Ans. 28000 dollars.

OPERATION.

Multiplicand

Multiplier

Product

700 40 28000

The multiplicand we resolve into the factors 7 and 100, and the multiplier into the factors 4 and 10. Now, it is evident (Art. 58), that, if these several factors be multiplied together, they will produce the same product as the origiThus 7 X 4 = 28, and 28 X 100: 28000, the same result as in the operation. Hence, when there are ciphers, one or more, on the right of the multiplier, or multiplicand, or both, we may, for the required product,

nal factors, 700 and 40.

Multiply the significant figures together, and to their product annex as many ciphers as there are on the right in both multiplicand and multiplier.

9. What must be the distance sailed by a steamship, whose average rate is 310 miles a day, in making a voyage from New York to Liverpool, in 12 days?

10. The annual salary of a member of Congress being 3,000 dollars, how much do 296 members receive?

Ans. 888,000 dollars.