11. The salary of the President of the United States is 25,000 dollars a year; how much will it amount to in 82 · years ? Ans. 2,050,000 dollars. 12. The earth is 95,000,000 of miles from the sun, and the planet Neptune is 30 times as far. How far is Neptune from the Sun ? Ans. 2,850,000,000 miles. DIVISION. be 66. Division is the process of finding how many times one number is contained in another; or the process of separating a number into a proposed number of equal parts. In division there are three principal terms: the Dividend, the Divisor, and the Quotient. The dividend is the number to be divided. The quotient is the result, or number produced by the division, and denotes the number of times the divisor is contained in the dividend, or one of the equal parts into which the dividend is divided. When the dividend does not contain the divisor an exact number of times, the excess is called a remainder, and may regarded as a fourth term in the division. When the dividend consists of a simple number, the process is termed Division of Simple Numbers. 67. Division is frequently indicated by writing the dividend above a short horizontal line and the divisor below; thus, The expression = 3 is read, 6 divided by 2 is equal to 3. Another method of indicating division, is by a curved line placed between the divisor and dividend. Thus, the expression 6 ) 12 shows that 12 is to be divided by 6. 68. When a number is divided into two equal parts, one of the parts is called one half; when divided into three equal parts, one of the parts is called one third, two of the parts two thirds ; when divided into four equal parts, one of the parts is called one fourth, two of the parts two fourths, three of the parts three fourths ; etc. Such equal parts are called FRACTIONS, since they are fractured or broken numbers. They are expressed by figures, in a form of division; thus, one half is written }; one third, }; two thirds, ; one fourth, £; two fourths, į ; three fourths, j; and may also be read, one divided by two, one divided by three, and so on. In any fraction, expressed in the manner now explained, the number above the line is called the numerator, and that below the line its denominator. Thus, in }, 1 is the numerator, and 2 the denominator. 69. When the divisor and dividend are of the same kind or denomination, the quotient will denote the number of times the divisor is contained in the dividend, and will be an abstract number. Thus, to find how many pencils at 6 cents each can be bought for 24 cents, we inquire how many times 6 cents are contained in 24 cents, which are 4 times. Hence, 4 pencils, at 6 cents each, can be bought for 24 cents. 70. When the divisor and dividend are not of the same kind or denomination, the divisor must be regarded as an abstract number, and will derote the number of equal parts into which the dividend is divided, and the quotient will denote the number of units in each part, and will be of the same kind or denomination as the dividend. Thus, to find the cost of 1 pencil, when 4 pencils cost 24 cents, we separate or divide the 24 cents into 4 equal parts, of which one part is 6 cents. Hence, 1 pencil costs 6 cents, when 4 pencils cost 24 cents. 71. The remainder will always be of the same kind or denomination as the dividend, since it is a part of the dividend. 72. Division is the reverse of multiplication. The dividend answers to the product, and the divisor and quotient to the factors, of multiplication. In multiplication, two factors are given, to find their product; and in division, one of two factors and their product are given, to find the other factor. 73. To divide simple numbers. Ex. 1. How many yards of cloth, at 4 dollars a yard, can be bought for 948 dollars ? Ans. 237 yards. OPERATION. OPERATION We first inquire how many Divisor 4 ) 9 48 Dividend. times 4, the divisor, is contained 2 3 7 Quotient. in 9, the first left-hand figure of the dividend, which is hundreds, and find it contained 2 times, and i hundred remaining. We write the 2 directly under 9, its dividend, for the hundreds' figure of the quotient. To 4, the next figure of the dividend, which is tens, we regard as prefixed the 1 hundred that was remaining, which equals 10 tens, and thus form 14 tens, in which we find the divisor 4 to be contained 3 times, and 2 tens remaining. We write the 3 for the tens' figure in the quotient, and the 2 tens that were remaining, equal 20 units, we regard as prefixed to 8, the last figure of the dividend, which is units, in which the divisor 4 is contained 7 times. Writing the 7 for the units' figure of the quotient, we have 237 as the entire quotient, equal the number of yards of cloth at 4 dollars a yard that can be bought for 948 dollars. 2. How many times does 3979 contain 17 ? Ans. 23414 times. We e say, 17 in 39, Dividend. 2 times. The 2 we Divisor 17 ) 3 9 7 9 ( 2 3 4 14 Quotient. write in the quo34 tient. 17 x 2 34, which we write 57 under the 39. 39 51 — 3455, to which bringing down the 69 next figure of the 68 dividend, we form 57. 17 in 57, 3 1 Remainder. times. The 3 we write in the quotient. 17 X 3 = 51, which we write under the 57. 57 - 51 = 6, to which bringing down the next figure of the dividend, we form 69. 17 in 69, 4 times. The 4 we write in the quotient. 17 X 4 - 68, which we write under the 69. 69 - 68 1, a remainder, or a part of the dividend left undivided. 1 divided by 17 = 1 (Art. 68). The i we write in the quotient, and obtain as the answer required 23417. In this illustration, to render the explanation the more concise, the naming of the denominations of the figures has been omitted. When, as in the operation preceding the last, results only are written down, the method is called short division ; and when, as in the last operation, the work is written out at length, it is called long division. The principle is the same in both cases. Hence the general RULE. — Beginning at the left, find how many times the divisor is contained in the fewest figures of the dividend that will contain it, for the first quotient figure. Multiply the divisor by this quotient figure, and subtract the product from the figures of the divirlend used. With the remainder, if any, unite the next figure of the dividend. Find how many times the divisor is contained in the number thus formed, and write the figure denoting the result at the right of the .former quotient figure. Thus proceed until all the figures of the dividend are divided NOTE 1. — The proper remainder is in all cases less than the divisor. If, in the course of the operation, it is at any time found to be as large as, or larger than, the divisor, it will show that there is an error in the work, and that the quotient figure should be increased. NOTE 2. — If at any time the divisor, multiplied by the quotient figure, produces a product larger than the part of the dividend used, it shows that the quotient figure is too large, and must be diminished. NOTE 3. — It will often happen that, when a figure of the dividend is taken, the number will not contain the divisor; and, in that case, a cipher must be placed in the quotient, and another figure of the dividend taken, and so on, until the number is large enough to contain the divisor. NOTE 4. - - If there be a remainder after dividing the last figure of the dividend, write it with the divisor underneath, with a line between them, at the right of the quotient. 74. First Method of Proof. — Multiply the divisor by the quotient, and to the product add the remainder, if any, and if the work be right, the sum thus obtained will be equal to the dividend. NOTE. — This method follows from division being the reverse of multiplica, tion. (Art. 72.) 75. Second Method of Proof. — Find the excess of nines in the divisor, quotient, and remainder. Multiply the excess of nines in the divisor and quotient together, and to the product add the excess of nines in the remainder. If the excess of nines in this sum equal the excess of nines in the dividend, the work may be supposed to be right. 76. Third Method of Proof. — Add together the remainder, if any, and all the products that have been produced by multiplying the divisor by the several quotient figures, and the result will be like the dividend, if the work be right. 77. Fourth Method of Proof.- Subtract the remainder, if any, from the dividend, and divide the difference by the quotient. The result will be like the original divisor, if the work be right. NotE. — The first method of proof (Art. 74) is usually most convenient, and is most commonly employed. PROOF BY THE NINES. Dividend. Divisor Divisor 759) 18988 ( 2 5 Quotient. Quotient 1 518 Remainder 3 excess. 7 excess. 4 excess. 3808 13 Remainder. (3 x 7) +4= 7 excess. Dividend 7 excess. Ans. 152434 5. Divide 147856 by 97. OPERATION PROOF BY ADDITION. 1 Dividend. +97 97 485 Products. 2 3 5 194 +194 388 416 28 Remainder. +388 1478 5 6 Dividend. Ans. 297. Dividend. Divisor 2 8 5) 8 4 6 4 5 ( 2 9 7 Quotient. 2 9 7) 8 4 6 45 ( 2 8 5 Divisor 570 594 2 7 64 2 5 2 4 2 5 6 5 2 3 7 6 1995 1485 1995 1485 PROOF BY DIVISION. |