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Multiplicand 218574693 | 12 times 3 are 36, carry 3. 12 times 9 and Multiplier.. 12 3 are 111, carry 11. 12 times 6 and 11 are

83, carry 8. 12 times 4 and 8 are 56, Product....2622896316 carry 5.

12 times 7 and 5 are 89, carry 8.

12 times 5 and 8 are 68, carry 6. 12 times 8 and 6 are 102, carry 10. 12 times 1 and 10 are 22, carry 2. 12 times 2 and 2 are 26. The product is 2,622;896,316.

Method of Proof. The most correct way of proving Multiplication is by Division; but this cannot be practised until Division is learned. It may also be proved when the multiplier is large, by changing the places of the multiplicand and multiplier.

1. Multiply 2. Multiply 3. Multiply 4. Multiply 5. Multiply 6. Multiply 7. Multiply 8. Multiply 9. Multiply 10. Multiply 11. Multiply

EXERCISES.
27 043 8 6 5
461937 54
5 3 7 29 16 8 4
1708 39 426
8 430716982
361809 25 4 7
972 36 5014

280 7 5 6 493
87 61340 259

740381692 4519 7 2 8 43 6

by by by by by by by by by

6. 7. 8. 9. 10. 11. 12.

by by

When the multiplier has two component parts that do not exceed 12.

Example. Multiply 31879645 by 72

9

286916805

8

Here 9 and 8 are the component parts of 72; because 9 times 8=72. First, multiply with one figure, as before taught; then multiply the product with the other.

Product 2295334440

The same Example proved.
31879645
12

Here 12 and 6 are also component parts

of 72, for 12 times 6=72. Hence the 382555740

product is 2,295;334,440, the same in both operations, which proves the work

right. Product 2295334440

Exercises.

12. Multiply 4729681 by 24. Product 113512344. 13. Multiply 5082763 by 35. Product 177896705. 14. Multiply 2368047 by 45. Product 106562115. 15. Multiply 824096 by 54. Product 44501184. 16. Multiply 107352 by 63. Product 6763176. 17. Multiply 635187 by 77. Product 48909399. 18. Multiply 1430268 by

84.

Product 120142512. 19. Multiply 5180376 by 108. Product 559480608. 20. Multiply 2173564 by 121. Product 263001244. 21. Multiply 8693408 by 132. Product 1147529856.

22. Multiply eight hundred and forty-nine thousand, three hundred and seventy five, by one hundred and forty-four.

Product 122310000.

When the multiplier consists of several figures, as in the following

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EXERCISES.

85.

23. Multiply 572638 by 47. Product 26913986. 24. Multiply 756024 by

Product 64262040. 25. Multiply 219738 by 93. Product 20435634. 26. Multiply 406523 by 107. Product 43497961. 27. Multiply 357269 by 235. Product 83958215. 28, Multiply 819605 by 573. Product 469633665. 29. Multiply 4076153 by 2085. Product 8498779005. 30. Multiply 2358746 by 7304. Product 17228280784. 31. Multiply 5072894 by 40809. Product 207019731245. 32. Multiply 7098356 by 93584. Product 664292547904.

33. Multiply one hundred and seventy-four thousand, and fiftyeight, by two hundred and eighty-five.

Product 49606530. 34. Multiply four millions; nine hundred and thirty-seven thousand, two hundred and sixty-eight, by three thousand and fifty-nine.

Product 15103102812.

35. Multiply seventy-four millions; and ninety-six thousand, three hundred and eighty-two, by forty-seven thousand, nine hundred and eight.

Product 3549809468856.

When there are ciphers at the right hand of the multiplicand, or multiplier, or both, as in the following

[blocks in formation]

Exercises.
36. Multiply 827460 by 90.
37. Multiply 398520 by 5300.

38. Multiply 716300 8500. 39. Multiply two hundred and forty-five thousand, by six thousand, nine hundred.

Product 1690500000.

by

In Multiplication, contractions are made use of, which shorten the work ;-thus

When the multiplier is more than 12, and less than 20, as in the following

Example. Multiplicand 928346

These multipliers (viz. 13, 14, 15, Multiplier.. 18

&c.) may be termed back-figures,

because in multiplying, the back, or 16710228 Product. figure behind the one used, is added

in. 8 times 6 are 48, carry 4;-8 times 4 and 4 are 36+the back figure 6=42, carry 4; 8 times 3 and 4 are 28+4=32, carry 3; 8 times 8 and 3 are 67+3=70, carry 7; 8 times 2 and 7 are 23+8=31, carry 3; 8 times 9 and 3 are 75+2=77, carry 7; 7 and 9 are 16. The Product is 16;710,228.

EXERCISES.
40. Multiply 2 3 5 7 6 8 by 13.
41. Multiply 4 7 29 35 by 15.
42. Multiply 138 709 by 17.
43. Multiply 80 6 7 2 5 by 19.
44. Multiply 5 7 2 8 0 9 6 by 16.

45. Multiply 9137208 by 18. Again, When the multiplier is 21, 31, 41, &c. to 121, as in the following

Example. Multiplicand 728496

These multipliers may be termed Multiplier.. 91

front-figures, because in multiply

ing, the figure in front of the one 66293136 Product. used, is added in. Bring down the

first figure of the multiplicand, viz. 6, then 9 times 6 are 54, and front-figure' 9=63, carry 6; 9 times 9 and 6 are 87+4–91, carry 9; 9 times and 9 are 45+8=53, carry 5; 9 times 8 and 5 are 77+2=79, carry 7; 9 times 2 and 7 are 25+7=32, carry 3; 9 times 7 and 3 are 66.

The product is 66;293,136.

EXERCISES.
46. Multiply 42 8 637 by 31.
47. Multiply 530 792 by 41.
48. Multiply 70 9 5 4 6 by 61.
49. Multiply 2753 89 by 81.
50. Multiply 1706 8 3 by 101.
51. Multiply 8 29 6 3 5 by 111.

52. Multiply 310 7 2 6 9 by 121.
Contractions used.

Common Method. 78 39 64

783964 1971

1971

5 5 6 61 444 1489 5 3 1 6

78 3 9 6 4 5 4 8 7 7 48 705 5 6 7 6 783 4

15 45 19 30 44

1 5 4 519 3 0 4 4

SIMPLE DIVISION

Teaches to find how often one number is contained in another of the same kind. When the divisor is not more than 12.

2 in 7, 3 times and 1 over, place 3 under Examples.

7 and carry 1; 1 considered as 10 and 4 Dividend.

are 14, 2 in 14, 7; 2 in 3, 1 and 1 over, Divisor 2) 7431284 carry 1; 1 considered as 10 and 1 are 11, Quotient.. 3715642

2 in 11, 5 and 1 over; 1 considered as 10 and 2 are 12, 2 in 12, 6; 2 in 8, 4;

2 in 4, 2. The quotient is 3;715,642. Dividend.

12 in 46, 3 and 10 over; 12 in 105, 8 Divisor 12) 465381793

and 9 over; 12 in 93, 7 and 9 over; 12

in 98, 8 and 2 over; 12 in 21, 1 and 9 Quotient... 38781816+1

over; 12 in 97, 8 and 1 over; 12 in 19, 1 and 7 over; 12 in 73, 6 and 1 over.

The quotient is 38;781,816+1 (plus 1). Method of Proof. Multiply the quotient by the divisor, add the remainder, if any, to the product, and if this product is the same as the Dividend, the work is right.*

* The first example in Division is a proof to the first example in Multiplication, see page 7.

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