EXAMPLES. 1. Divide 695.57270875 by 52.35775, and retain in the quotient three places of decimals. FIRST OPERATION. 52.3 5 7 7 5) 6 9 5.5 7 2 7 0 8 7 5 (1 3.2 8 5. 5 2 3 5 8 product by 1, +1. 17199 Ans. 13.285. By inspection, it is evident that the first quotient figure will be of the order of tens, and there 1492 fore the quotient will contain two places of whole 445 numbers; and as 419 = 26 three places of decimals, it must product by 8, +3. there are to be 26 product by 5, +1. contain five fig SECOND OPERATION. 5 2.3 5 7 7 5) 69 5.5 7270875 (13.2 85. 5 2 3 5 7 7 5 171995 20 149 21958 4450 4087 26178875 duct by having regard to rejected figures, as in contracted multiplica tion of decimals (Art. 273). The nature and extent of the contraction will be seen by comparison with the common method as shown in the second operation, in which the vertical line cuts off the figures not required. NOTE. When the given divisor does not contain as many figures as are required in the quotient, we must begin the division in the usual way, and continue till the deficiency is made up, after which begin the contraction. 2. Divide 4327.56284563 by 873.469, and retain five decimal places in the quotient. Ans. 4.95445. 3. Divide 252070.520751 by 591.57, and terminate the operation with four decimal places in the quotient. Ans. 426.1043. 4. Divide 70.23 by 7.9863, and retain in the answer four decimals. 5 Divide 12193263.1112635269 by 1234.56789, and let the quotient contain as many decimal places, plus one, as there will be integers in it. Ans. 9876.54321. REDUCTION OF DECIMALS. 277. To reduce a decimal to a common fraction. Ex. 1. Reduce .125 to its equivalent common fraction. OPERATION. .125= +1250 = 200 = 7% = Ans. Ans.. Erasing the decimal point and supplying the denominator, which is understood, we have 125, which reduced to its lowest terms equals , the answer required. RULE. Erase the decimal point, and write under the numerator its decimal denominator, and reduce the fraction to its lowest terms. EXAMPLES. 2. Reduce .875 to a common fraction. Ans.. Ans. 18. 3. Reduce .9375 to a common fraction. Ans. 5. Change .00075 to the form of a common fraction. 6. Express 31.75 by an integer and a common fraction. Ans. 31. 7. Express 96.024 by an integer and a common fraction. Ans. 96. 8. Express 163.04 by an integer and a common fraction. 9. Express 1001.4375 by an integer and a common fraction. Ans. 1001 10. Express 1457.222 by an integer and a common fraction. 11. Express 19678.36 by an integer and a common fraction. 12. Express 9163.8755 by an integer and a common fraction. Ans. 91631731. 278. To reduce a common fraction to a decimal. Ex. 1. Reduce to a decimal. Ans. .375. OPERATION. 8) 3.0 (3 tenths. 24 8) 60 (7 hundredths. 56 8) 40 (5 thousandths. 40 Ans. .375. Or thus: .37 5 Ans. Since we cannot divide the numerator, 3, by 8, we reduce it to tenths by annexing a cipher, and then dividing, we obtain 3 tenths and a remainder of 6 tenths. Reducing this remainder to hundredths by annexing a cipher, and dividing, we obtain 7 hundredths and a remainder of 4 hundredths; which being reduced to thousandths by annexing a cipher, and then divided, gives a quotient of 5 thousandths. The sum of the several quotients, .375, is the answer. To prove that .375 is equal to 8, we change it to the form of a common fraction, by writing its denomi nator, and reducing it to its lowest terms. Thus, .375 375 = 1000 RULE. Annex ciphers to the numerator, and divide by the denominator. Point off in the quotient as many decimal places as there have been ciphers annexed. NOTE. It is not usually necessary that the decimals should be carried to more than six places. When a decimal does not terminate, the sign plus (+) is generally annexed. Thus, in the expression .333+, the sign annexed indicates that the division could be carried further. 6. Reduce 19 to an equivalent decimal expression. Ans. $315.875. 8. Reduce $11632 to an equivalent decimal expression. Ans. 1163.75. NOTE. A decimal with a common fraction annexed constitutes what is called a complex decimal; as, .871, .314, and .183. In such expressions, instead of the common fraction, its equivalent decimal, with the decimal point omitted, may be substituted. Thus, .4 .404. 9. Reduce .62 to a simple decimal. 10. Reduce .37 to a simple decimal. Ans. .625. Ans. .370625. 11. Reduce $4.314 to a simple decimal expression. Ans. $4.3125. 12. Reduce $ 60.183 to a simple decimal expression. Ans. $60.1875. 13. What decimal expression is equivalent to 14. What decimal expression is equivalent to +of of 4,- 1.05? 23, + 0.37₫, Ans. 2.9875. 279. To reduce a simple or compound number to a decimal of a higher denomination. Ex. 1. Reduce 15s. 9d. 3far. to the decimal of a £. OPERATION. 4 3.00 12 9.7 500 far. d. 2 0 1 5.8 1 250 s. .790 625 £. Ans. .790625. We commence with the 3far., which we reduce to hundredths by annexing two ciphers; and then, to reduce these to the decimal of a penny, we divide by 4far., since there will be as many hundredths of a penny as of a farthing, and obtain .75d. Annexing this to the 9d., we divide by 12d., since there will be as many shillings as pence; and then, the 15s. and this quotient by 20s., since there will be as many pounds as shillings, and obtain Hence the following .790625£. for the answer. RULE. Divide the lowest denomination, annexing ciphers if necessary, by that number which will reduce it to one of the next higher denomination. Then divide as before, and so continue dividing till the decimal is of the denomination required NOTE 1. The given number may also be first reduced to a common fraction of the given denomination (Art. 256), and then the fraction changed to a decimal. Thus, if it be required to reduce 15s. 6d. to a decimal of a £.; 15s. 6d. = 186d.; 1£. = 280d.; 188 £. = }} £. : .775 £. Answer. NOTE 2. ·Shillings, pence, and farthings may be readily reduced to a decimal of three places, by inspection, thus: Call half of the greatest even number of shillings TENTHS, and, if there be an odd shilling, call it 5 HUNDREDTHs; reduce the pence and farthings to farthings, and increase them by 1, if they amount to 24 or more, for THOUSANDTHS. Thus, if it be required to reduce, by inspection, 19s. 10d. 2far. to the decimal of a £.; half of 18s. 9s., which denote a value of .9£.; the 1s. denotes a value of .05£.; and 10d. 2far. 42far., which increased by 1far. = 43far., which denote a value of .043£.; .9£. + ,05£..043£. == .993£. Answer. = The reason for this process is, that 2s. equal a tenth of a £.; 1 shilling equals 5 hundredths of a £., and 1 farthing equals £., or so nearly a thousandth of a £. that 24 farthings exactly equal 25 thousandths of a £.; and therefore farthings require to be increased only by 1 when they amount to 24 or more, to denote with sufficient accuracy their value in thousandths of a £. EXAMPLES. 2. Reduce 9s. to the fraction of a pound. Ans. .45, 3. Reduce 15cwt. 3qr. 14lb. to the decimal of a ton. 4. Reduce 2qr. 21lb. 8oz. 12dr. to the decimal of a cwt. Ans. .71546875. 5. Reduce 1qr. 3na. to the decimal of a yard. Ans. .4375. 6. Reduce 5fur. 35rd. 2yd. 2ft. 9in. to the decimal of a mile. Ans. .73603219+. 7. Reduce 3gal. 2qt. 1pt. of wine to the decimal of a hogshead. Ans. .0575396+. 8. Reduce 1pt. to the decimal of a bushel. Ans. .015625. 9. Reduce 2R. 16p. to the decimal of an acre. Ans. .6. 10. Reduce 175 cubic feet to the decimal of a ton of timber. Ans. 4.375. 11. Reduce 3.755 pecks to the decimal of a bushel. Ans. .93875. 12. What decimal part of a degree is 25′ 34′′.6? 13. Reduce 12T. 3cwt. 2qr. 20lb. to hundred-weight and the decimal of a hundred-weight. Ans. 243.7. 14. Reduce 2hhd. 30gal. 2qt. 14pt. to gallons and the decimal Ans. 156.6875. of a gallon. 15. Reduce to the decimal of a pound, 19s. 113d., 16s. 91d., and 17s. 51⁄2d., and find their sum. Ans. 2.710416+. 280. To find the value of a decimal in whole numbers of lower denominations. Ex. 1. What is the value of .790625 £.? OPERATION. .790 625£. 1 5.8 1 2 5 0 0s. 9.7 5 0 0 0 Od. 4 3.0 0 0 0 0 Ofar. Ans. 15s. 9d. 3far. Ans. 15s. 9d. 3far. There will be 20 times as many millionths of a shilling as of a pound; therefore, we multiply the decimal, .790625, by 20, and reduce the improper fraction to a mixed number by pointing off six figures on the right, which is dividing by its denominator, 1000000. The figures on the left of the point are shillings, and those on the right, the decimal of a shilling. The decimal .812500 we multiply by 12, and, pointing off as before, obtain 9d., and a decimal of a penny. The decimal |