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What is the product of — 3. Of 80276 x 39 ?

13. Of 30678 x 427 ? 4. Of 298794 X 148 ? 14. Of 5.796 X 238 ? 5. Of 273986 X 27 ?

15. Of 45.718 X 432 ? 6. Of 4943 X 78?

16. Of .00876 x 74 ? 7. Of 23879 X 2741 ? 17. Of 67.968 X 327 ? 8. Of 84976 X 203 ? 18. Of 970062 x 37 ? 9. Of 25873 x 506 ? 19. Of 29743 X 806 ? 10. Of 47.296 x 37 ?

20. Of 8427 X 3076 ? 11. Of 423758 x 6200 ? 21. Of 279437 X 27623 ? 12. Of 594.27 x 20200 ? 22. Of 8.64298 X 43000 ?

(f.) It is obvious that the product obtained by multiplying one number by the difference of two numbers, is equivalent to the difference of the products obtained by multiplying the numbers separately by the two numbers.

Illustration. - 5 times 3 = 7 times 3 – 2 times 3= 8 times 3 times 3 = 15 times 3 · 10 times 3 29 times 3 24 times 3, &c.

(9.) This principle is the reverse of that stated in (a,) and can often be advantageously applied, as illustrated below.

(h.) To multiply by 99, observe that since 99 = 100 — 1, 99 times a sumber must equal 100 times the number minus once the number. For example,

99 times 837 = 100 times 837 - 837 = 83700 837 - 82863.

(i.) To multiply by 999, observe that since 999 = 1000 - 1, 999 times a number must equal 1000 times the number minus once the number. For example,

999 times 14.67 = 1000 times 14.67 14.67 = 14670 - 14.67 = 14655.33. (j.) To multiply by 699, observe that since 699 = 700

· 1, 699 times a number must equal 700 times the number - once the number For example,

699 x 5784 = 700 X 5784 5784 = 4048800 5784 = 4013016 (k.) In like manner we should have 49 X 785 = 50 X 785 - 785. 98 X 4697 = 100 X 4697 – 2 X 4697. 79996 X 394845 80000 X 394845 4 X 394845. What is the product 23. Of 48673 X 29 ? | 24. Of 37848 X 99 ?

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82. Abbreviated Method. (a.) When the multiplier consists of more than one denomination, much labor in writing figures may be saved by applying the principles illustrated in the following examples: —

1. What is the product of 8356 multiplied by 79 ?

Preliminary Explanation. — It is evident that, in performing the required multiplication, we shall obtain units by multiplying 6 units by 9 units. We shall obtain tens by multiplying 5 tens by 9, and 6 units by 7 tens, and we may have some from the product of the units. We shall obtain hundreds by multiplying 3 hundreds by 9, and 5 tens by 7 tens, and we may have some from the former products. We shall obtain the other denominations in a similar manner. We may then proceed thus, writing the figures of each denomination as usual:

WRITTEN WORK.
8356 Multiplicand.

79 Multiplier.

660124 Product. Explanation. 9 times 6 units = 54 units. We write 4 units.

9 times 5 tens = 45 tens, + 5 tens (from the product of the units) = 50 tens, + 7 tens times 6, or 42 tens, = 92 tens = 9 hundreds and 2 tens. We write 2 tens.

9 times 3 hundreds = 27 hundreds, + 9 hundreds (from the previous product) = 36 hundreds, + 7 tens times 5 tens, or 35 hundreds, = 71 hundreds = 7 thousand and 1 hundred. We write 1 hundred.

9 times 8 thousands = 72 thousands, +7 thousands (from the previous product) = 79 thousands, + 7 tens times 3 hundreds, or 21 thousands, = 100 thousands = 10 ten-thousands. We write 0 in the thou. sands' place of the product.

7 tens times 8 thousands = 56 ten-thousands, + 10 ten-thousands (from the previous product) = 66 ten-thousands, which we write.

The multiplication being now completed shows the answer to be 660124.

(6.) The following exhibits the necessary operations on the numbers, and is practically a much more convenient solution than the full form above given.

9 X 6= 54 units. 5 (from last product) + 9 X 5, +7 X 6=5+ 45 + 42 = 92 tens.

9 (from last product) + 9 X 3, +7 X 5 =9+ 27 + 35= 71 hundreds.

7 (from last product) + 9 X 8, + 7 X 3 = 7+ 72 + 21 = 100 thousands.

10 (from last product) + 7 X 8= 10 + 56 = 66 ten-thousands. This gives for an answer 660124, as did the first method.

(c.) The last process being understood, the work may be still further abbreviated by omitting to name the factors used.

Thus, 54 units = 5 tens and 4 units.
45 +5= 50, + 42 = 92. 92 tens = 9 hundreds and 2 tens.

27 +9= 36, + 35 = 71. 71 hundreds = 7 hundreds and I thou. sand.

72 +7=79, + 21 = 100. 100 thousands = 10 ten-thousands and 0 thousands.

56 + 10 = 66. 66 ten-thousands. Answer, as before, 660124.

(d. Finally, the work may be abbreviated so as to name only results :

54 units = 5 tens and 4 units.
45, 50, 92 tens = 9 hundreds and 3 tens.
27, 36, 71 hundreds = 7 thousands and 1 hundred.
72, 79, 100 thousands = 10 ten-thousands and 0 thousands.
56, 66 ten-thousands.

Note. — The above methods are much more expeditious than is the method of writing the product by each figure of the multiplier separate. ly, and are no more liable to inaccuracy.

2. How much will 97 acres of land cost at $347 per acre?

3 If a cubic yard of sand weighs 2537 lb., how much will 38 cubic yards weigh?

4. How many pounds are there in 18 T. 17 cwt. 1 qr. ?

5. If a ship sails 96 miles in one day, how far will she sail in 247 days ?

6. Bought 24 bundles of hay, each bundle containing 197 lb. How many pounds were there in all ?

7. Bought 2947 gallons of oil at $.84 per gallon. How much did it cost ? Sold it for $.97 per gallon. How much was received for it? What was the gain on it ?

8. Mr. Russell bought 86 balls of twine, each ball containing 8794 ft., and Mr. Greene bought 57 times as much. How many feet of twine did Mr. Russell buy? How many did Mr. Greene buy?

9. How much will 83 casks of old wine cost at $138.47 per cask ?

10. How much will 67 tons of lead cost at $139.48 per ton ?

11. Mr. Hovey bought 6247 feet of land, and Mr. Ewell bought 94 times as much. How many feet did Mr. Ewell buy ?

12. How many pounds are there in 958 boxes of sugar, each box containing 743.67 lb. ?

NOTE. — The products and sums employed in solving the above example are given below, but the pupil should be prepared to give a more full explanation.

56 hundredths = 5 tenths and 6 hundredths. 5 + 48 + 35 = 88. 88 tenths = 8 units and 8 tenths. 8 + 24 + 30 + 63 = 125. 125 units = 12 tens and 5 anits. 12 + 32 + 15 + 54 = 113. 113 tens = 11 hundreds and 3 tens.

11 + 56+ 20 + 27 = 114. 114 hundreds = 11 thousands and 4 hundreds.

11 + 35 + 36 = 82. 82 thousands = 8 ten-thousands and 2 thousands.

8 + 63 = 71. 71 ten-thousands.
The answer, therefore, is 712435.86 lb.
13. How many are 8795 times 96543 ?

Note. — By extending the principles before explained, we can write the final product at once, as below.

96543
8795

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In the following forms, two products are written :

96543
8795

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14. If a cubic foot of iron weighs 486.25 lb., how much will 347 cubic ft. weigh?

15. How many square feet are there in a rectangular lot, 4327 feet long and 249 feet wide ? "

16. How much will 48 acres of land cost at $23.968 per acre ?

17. What will 798 tons of hay cost at $14.278 per ton ?

18. I bought 287 bales of cloth, each bale containing 247.986 yards. How many yards did they all contain ?

19. How many square inches are there in a lot 247 ft. long and 187 ft. wide ?

20. What will 47983 yards of cloth cost at $2.83 per yd. ? 21. What will 7894 bbl. of flour cost at $6.37 per bbl. ?

22. How many solid inches in 5 C. 6 Cd. ft. 12 cu. ft. 1437 cu. in. ?

23. How many dr. in 18 T. 16 cwt. 1 qr. 14 lb. 6 oz. 11 dr.?

24. A grain dealer sold 287 bushels of wheat at $1.294 per bushel, and 1479 bushels at $1.267 per bushel. What did he receive for it ?

25. I bought 48 yards of broadcloth at $3.875 per yd., 153 yards of doeskin at $1.166 per yd., and 379 yards of cassimere at $1.125 per yd. What was the cost of the whole?

26. Mr. Aldrich owns 4 house lots, the first 328 ft. long and 189 ft. wide; the second 437 ft. long and 249 ft. wide;

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