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. 2 and 2 are 26. The product is 2,622;896,316.

12 times

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.

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

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

39. Multiply two hundred and forty-five thousand, by six thousand, nine hundred. Product 1690500000.

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
Multiplier.. 18

16710228 Product.

These multipliers (viz. 13, 14, 15, &c.) may be termed back-figures, because in multiplying, the back, or 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.

40. Multiply

41. Multiply

EXERCISES.

235768 by 13.
472935 by 15.

42. Multiply 13870 by 17.
43. Multiply 806725 by 19.
44. Multiply 5728096 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

Multiplier.. 91

66293136 Product.

These multipliers may be termed front-figures, because in multiplying, the figure in front of the one 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 4 and 9 are 45+8=53,

carry 5; 9 times 8 and are 77+2=79, carry 7; 9 times are 25+7=32, carry 3; 9 times 7 and 3 are 66.

and 7

The product is 66;293,136.

Teaches to find how often one number is contained in another of the same kind.

When the divisor is not more than 12.

Examples.
Dividend.

Divisor 2) 7431284
Quotient.. 3715642

Dividend.

Divisor 12) 465381793

Quotient... 38781816+1

2 in 7, 3 times and 1 over, place 3 under
7 and carry 1; 1 considered as 10 and 4
are 14, 2 in 14, 7; 2 in 3, 1 and 1 over,
carry 1; 1 considered as 10 and 1 are 11,
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.
12 in 46, 3 and 10 over; 12 in 105, 8
and 9 over; 12 in 93, 7 and 9 over; 12
in 98, 8 and 2 over; 12 in 21, 1 and 9
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.