x = quotient. Therefore to divide a fraction by a whole number, multiply the denominator by that number, except it will divide the numerator, as in Ex. 4. 58. If the divisor is a whole, and the dividend a large mixt number, divide the parts separately, and then add the quotients together. Ex. 10. Required the 5th. part of 45614124? 5) 4561412 9122822 the integral part divided by 5. the fraction 3 divided by 5. Sum 91228217 the answer. 59. When the divisor is a small fraction and the dividend a a large mixt number, multiply the latter (without reducing it to an improper fraction) by the denominator of the divisor, and divide the product by the numerator. 60. In like manner the quotient is found in the contracted method of division of whole numbers when the divisor is the product of two or more factors. (30. Ex. 11.) Ex. 12. Let 8783 be divided by 56, or 7 times 8. OF DECIMALS. 61. DECIMALS are Fractions in the form of whole numbers, but whose values decrease from the place of units progressively to the right hand in the same decuple or tenfold proportion as the common scale of whole numbers increase to the left. They are usually separated from the integers by a comma or dot, the decimals being on the right hand. when the fraction is set down. Thus the mixt number 21, decimally will be 21.2; the 2 on the right of the 1, or dot, denotes 2 tenths, whereas the other 2 on the left are 2 tens. Another 2 on the left will be 2 hundreds, but on the right 2 hundredths, (3), and the whole or 221 22 is 221, because and together make. A third figure on the left will be thousands, but on the right, the like number of thousandth parts. Thus 5008-005 is the same as 5008; and 5000.005 the same as 5000 Consequently a decimal fraction has always either 10, 100, 1000, &c. for its denomi nator; viz. the number of equal parts into which the integer or whole is supposed to be divided. For example, let a foot in length be the integer, and conceive it to be divided into 100 equal parts; then 25 (or 25 with a dot on the left) will be the decimal part of a foot denoting 3 inches or ( being (% = 4). And 1 inches or of a foot will be 125 of a foot, because 125 is of 1000; the foot in this case is supposed to be divided into 1000 equal parts. Therefore to read, or set down a proposed decimal, it is only necessary to remember that the denominator is 1 with as many ciphers annexed as there are decimal places, or that the same number of figures to the right of the decimal point have always the same common denominator. Thus the denominator of the fractions 5000, 0746, 0005, is 10000. that the value of a decimal fraction is not And hence it altered by ciphers on the right hand; for 5000 (or 6) when reduced to its lowest terms is the same as 5, each being equal to . ADDITION AND SUBTRACTION OF DECIMALS. 62. PLACE the numbers so that the decimal points may stand directly under each other; then add, and subtract, as in whole numbers, and set the decimal point in the sum or difference directly under the points above. Ex. 1. Required the sum of 7, 014, and ⚫1246 ? *014 Sum .8386 By placing the decimal points under each other, tenths are brought under tenths, hundredths under hundreths, &c. whence the method of addition becomes the same as that for whole numbers. The decimals in the foregoing example set down as vulgar fractions are, and 1246 and when brought to a common denominator will be 788, 160, and 1246 Τουσσ 10000 hence 7000 140 1246 10000 8386 the sum of the numerators, and the sum of the fractions as before: but this is evidently nothing more than reducing the decinals to a common denominator by annexing ciphers on the right hand : Thus 7000 Sum 8386 Ex. 2. What is the sum of 0159, 54.77 and 9.299? ⚫0159 54.77 9.299 Sum 64:0849 Ex. 3. Required the sum of 9 tenths, 19 hundredths, 18 thousandths, 211 hundred thousandths, and 19 millionth parts? .9 •19 ⚫018 ⚫00211 ⚫000019 Sum 1110129 4. Required the difference of 406 and 11? •406 •296 Ans. 5. What is the difference of 49.01 and 9078 ? 49.01 48 1022 Ans. 6. What is the difference of 1 and 24g 0.042 0.958 Ans, 7. Required the difference of 594-0012 and 24.98 ? 594-0012 569 0212 Ans, MULTIPLICATION OF DECIMALS. 63. MULTIPLY as in whole numbers, and point off as many places for decimals in the product as there are decimals in both multiplier and multiplicand; but if there should not be so many, put ciphers on the left to supply the defect. Ex. 1. Required The product of 2 and 03 ? *03 •2 The decimals 2 and 03 when set down as vulgar fractions will be 6 7000 and, and their product = 1ooo or 6 thousandth parts, as before. Hence the truth of the rule is evident. Product 6420 000. Therefore multiplying by 10, 100, 1000, &c. is only removing the decimal point so many places to the right as there are ciphers in the multiplier. Thus $2.1 multiplied by 10 is 821; 44 multiplied by 1000 is 4400, &c. 64. There is a method of contracting the operation so as to retain only a proposed number of decimals in the product. Let 5849 be multiplied by 7.26, and the product have only 3 decimal places. ⚫5849 35/994 11698 4094 3 4-24 13 4 The required product is 4.246. But to omit setting down the figures on the right of the perpendicular bar, yet retain the product to the left, it is evident that the multiplication by the integer 7 must begin at 4 in the multiplicand or the 3d. place in the decimal from the left (3 being the number of decimals to be retained); the multiplication by 2 must begin at the 8; and that by 6 at the 5, remembering to carry from the figures omitted on the right hand, as in common multiplication. But when the figures of the multiplier are set down |