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at the left of the 2 in the product, and the work is done. I then prove it by casting out the nines, &c. as above directed.
CASE II. When the multiplier exceeds 12.
RULE. Place the numbers as directed in Case I, then multiply by each figure in the multiplier separately as in that case, placing the first figure of each product under its multiplier ; then add the several producis into one sum, which will be the true product.
CASE III. When there are ciphers between the significant figures of the multiplier, omit them, and multiply by the significant figures only, observing the directions in Case II, to place the first figure of each product under its multiplier.
7 3 6 5 9 1 Multiplicand.
5 8 9 2 7 2 8
2 9 4 6 3 6 4 4 4 1 9 5 4 6 2 2 0 9 7 7 3
2 6 5 2 0 8 1 1 6 3 6 8 Product.
Products. 2 27846375 63400006 1765460342078250 3 5968 7069
42329172 7369 37004
272682476 5 74965 600067
44984022655 6 96384 730436
70402343424 CASE IV. When ciphers are annexed to either, or to both factors, neglect them, and proceed with the significant figures as before directed, and to the product annex as many ciphers as there are in both factors.
2 X 5 Proof. 2 1 8 0 5
2 3 4 5 6 7 8
24. 4 2 1 60000 Product.
369470 4736900 85371000
43620000 3473280000 282000000
CASE V. When the multiplier (exceeding 12) can be found in the multiplication table, take the two numbers or figures which produce it, and multiply by one of those figures or numbers, and then multiply this product by the other, the second will be the true product.
SUPPLEMENT TO SIMPLE MULTIPLICATION.
CASE I. When the multiplier is a fraction, as , &c. multiply by the numerator of the fraction, and divide the product by the denominator ; the quotient will be the true product : If the numerator be 1, divide by the denominator only, as I will not multiply.
NOTE. A fraction is a part of an unit or whole number, and is expressed by two numbers placed one over another, thus, is read three fourths, sis seven eighths, &c.
CASE II. When the multiplier is a mixed number, as, 5, 239, &c.
Multiply the multiplicand by the numerator of the fraction, divide the product by the denominator ; then multiply the multiplicand by the whole number of the multiplier; add this product and the quotient together, their sum will be the true product.
CASE III. When the multiplicand is a mixed number, multiply the multiplier by the numerator of the fraction ; divide the product by the denominator ; then proceed as in CASE 11, with the whole numbers, &c.
By 36 11
Teaches to find how often one number is contained in another of the same denomination.
The number to be divided is called the dividend.
That by which it is to be divided is called the divisor,
The number found by the operation is called the quotient.
The remainder, if there be any, will be less than the divisor.
When the divisor does not exceed 12, it is called Short Division.
In Short Division, this is the Rule ;
Set the divisor at the left of the dividend, with a separating line between them ; draw a line under the dividend.
Inquire how often the divisor is contained in the first figure (or as many as necessary) of the dividend, set the resulting figure under the right hand figure divided, if more than one is divided.
If there be a remainder, conceive it prefixed to the next dividend figure, which divide as before, and so proceed till the whole is divided, setting each resulting figure to the right of the last.
When a remainder is conceived to be prefixed to the next dividend figure, and the divisor is not contained in this increased number, set 0 under it in the quotient and then conceive the next dividend figure to be annexed to the last increased number; if the divisor is not then contained, set another 0 under it, and conceive the next dividend annexed, and so on till the divisor is contained in the increased number, or the whole of the divi. dend figures annexed ; if, when the last dividend figure is used, the divisor is not contained, set 0 under it, and then all the dividend figures used, increased by one or more prefixed, will be the true remainder.
To prove Division, multiply the quotient by the divisor ; to the product add the remainder, if any, sult will be equal to the dividend, if the work be right.