When there are not so many figures in the product as there are decimals in the factors, prefix a cipher or ciphers, until there be as many figures in the product as there are decimals in the factors, and point off as before directed. 9. Multiply six and one fourth cents by the same. Ans. ,00390625. A composite number is one which can be measured exactly by another number exceeding unity. Thus, 6 may be divided by 3 or 2. Hence, 2 and 3 are the component parts of 6. The number 15 may be divided by 3 or 5; therefore, 3 and 5 are the component parts of 15. When the multiplier is a composite number, the operation may be made easier, by multiplying first by one component part of the multiplier, and that product by the other. 10. Multiply 89 by 49. Ans. 4361. When the multiplier is 1, with any number of ciphers annexed, the multiplication may be performed by annexing as many ciphers to the multiplicand as there are ciphers in the multiplier. 11. Multiply 89 by 100. 12. Multiply 1728 by 100000. Ans. 8900. Ans. 172800000. Multiplication may be proved by casting out the nines, or threes, out of the multiplicand, multiplier, and product. EXAMPLE. Multiply 17249 by 1725 17249 1725 86245 34498 120743 17249 29754525 Having performed the multiplication, begin with the multiplicand, and add the figures from the right hand to the left, dropping the nines, and place the excess at the right hand of the cross. Cast out the nines of the multiplier, and place the excess at the left hand of the cross. Multiply these two figures together, cast out the nines of their product, and place the excess over the product. Finally, cast out the nines of the product, and place the excess under the cross; and if the figures at the top and bottom agree, the work is supposed to be right. Multiplication may be proved by division, addition, and subtraction. To prove multiplication by division, divide the product by the multiplier, and the quotient will be the multiplicand. To prove multiplication by addition, set down the multiplicand as many times as there are units in the multiplier, and add their several numbers, and their amount will agree with the required product. If the multiplier be a fraction, set down such a part of the multiplicand as is denoted by the multiplier. To prove multiplication by subtraction, subtract the multiplicand from the product as many times as there are units in the multiplier; if the work is right, there will be no remainder. If the multiplier be a fraction, take such a part of the multiplicand from the product as is denoted by the multiplier. EXAMPLES. 8X4−3. Proof by division. 32-4-8. Proof by subtraction. Hence it is obvious that multiplication is a method of repeating a given number a certain number of times. This repetition may be made by successive additions, and may therefore be proved by addition and subtraction, as well as division, and casting out the nines. VI. Division. Division consists in finding one factor, when the other, with the product, is given. Multiplication may be performed, as we have seen, by successive additions; in like manner, division may be performed by repeated subtractions of the divisor from the dividend. The number to be divided is called the dividend, the given factor is called the divisor, and the factor sought, is called the quotient. SIGN. The sign of division is a short, horizontal line between two dots, thus,÷, and shows that the number at the left hand of the line is to be divided by the number at the right hand; thus, 8÷4-2. RULE. Write the divisor at the right or left hand of the dividend, and draw a vertical line between it and the dividend. Seek how many times the divisor is contained in the first left-hand figure, or figures, of the dividend. Write the figure, showing how many times the divisor is contained in the first left-hand figure, or figures, of the dividend, on the right-hand side of the dividend, under the divisor, calling it the quotient figure. Multiply the divisor by the quotient figure, and place the product under the left-hand figure, or figures, of the dividend in which the divisor was contained once, or more. Subtract this product from the figures above it, and to the right hand of the remainder bring down the next undivided figure of the dividend, and proceed as before directed. Proceed in this manner till all the figures of the . dividend be divided. If there be a remainder, ciphers may be added, and the division be continued to an indefinite extent, or till there be no remainder. Point off as many figures for decimals from the right hand of the quotient as the decimals in the dividend exceed those in the divisor; that is, the decimals in the divisor and quotient must equal the number of decimals in the dividend. 2. Divide 738652 by 9. 8. Divide ,5 by ,000005. Ans. 82072, and 4 rem. Ans. 10487. Ans. 34. Ans. 6058,4. Ans. 2,4. Ans. 1000. Ans. 100000. When the divisor consists of one or more figures with ciphers annexed, cut off the ciphers and as many figures on the right hand of the dividend, and proceed with the remaining figures, as in the foregoing examples. The figures cut off in the dividend must be annexed to the remainder. Ans. 17530. 9. Divide 8765256 by 500. Rem. 256. When the divisor consists of only one figure, or of a number less than 12, a part of the process may be carried on in the mind, and the process shortened. 10. Divide 7956 by 6. 67956 1326 quotient. In this example I find that 6 are contained in 7 once, and there is 1 remainder. This 1 remainder, being a part of 7, retains the local value of 7, or thousands, and when taken with 9 hundreds, makes 19 hundreds. I then seek how many times 6 are contained in 19; I find the quotient figure to be 3, and there is 1 remainder. This 1 remainder is a part of the 19 hundreds, and retains the local value of hundreds, and, when taken with 5 tens, makes 15 tens. I then seek how many times 6 are contained in 15; and find the quotient to be 2, and there are 3 remainder. This figure, 3, retains the value of tens, and, when taken with 6 units, makes 36 units. I then find that 6 are contained in 36 6 times. 11. Divide 738652 by 9. Ans. 82072, and 4 rem. When the divisor is a composite number, the process may be varied by dividing first by one component part, and that quotient by the other. When the divisor is 10, 100, 1000, &c., cut off from the right hand of the dividend as many figures as there are ciphers in the divisor; the figures on the left of the point will be the quotient, and the figures on the right of the point will be the remainder. 16. Divide 8756 by 100. 17. Divide 54321 by 1000. Ans. 87,56, Ans. 54,321. To prove division, multiply the quotient by the divisor, and to that product add the remainder, if there be any; and if the work is right, the product will agree with the dividend. VII. Fractions.* Fractions are of two kinds, VULGAR and DECIMAL. The rules for Decimals will be given in another part of this work; therefore, it will be necessary, in this place, to consider only VULGAR FRACTIONS. is A fraction is a part of a unit. Thus, 1 is a unit; a part of 1, and is, therefore, a fraction. Fractions arise from division. If we divide 13 by 5, thus, 5|13|23, the quotient will be 2, and the remainder 3. This remainder is an undivided part of the dividend 13; and, as 5 are not contained in 3, the division of 3 may be expressed by writ • This article on FRACTIONS was prepared wholly by the Editor. |