EXAMPLES. FIRST OPERATION. SECOND OPERATION. given di 1. Divide 695.57270875 by 52.35775, and retain in the quotient three places of decimals. Ans. 13.285. By inspection, 5 2.3 5 7 7 5 ) 6 9 5.5 7 2 7 0 8 7 5 (1 3.2 8 5. it is evident that 5 2 3 5 8 = product by 1, + 1. the first quotient figure will be 1 7 1 9 9 of the order of 15 7 0 7 = product by 3, + 2. tens, and there fore the quotient 1492 will contain tuo 1 0 47 = product by 2, + 1. places of whole 445 numbers; and as 419 product by 8, +3. there are to be three places of 2 6 decimals, it must 26 = product by 5, + 1. contain five fig ures. Hence, we divide at first by 5 2.3 5 7 7 5 ) 69 5.5 7 2 7 0 8 7 5 (1 3.2 8 5. five figures of the 5 2 3 5 7.7 5 visor, counting 1719 9 5 2 0 them from the 1 5 7 0 73 2 5 left toward the right, thus using 149 21 9 5 8 the 52.357 and 10 471 5 5 0 rejecting the figures, 75, on 4 4 50 4 0 87 the right. In 41 88 6 2 0 0 multiplying each contracted divi2 6 1 7 8 8 7 5 sor by its quo2 6 1 7 8 8 7 5 tient figure we increase the product by having regard to rejected figures, as in contracted multiplication 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. Ans. š. OPERATION. .125 = 72 = 36 = = $ Ans. Erasing the decimal point and supplying the denominator, which is understood, we have 1367, 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 Ans. š. Ans. 1 2. Reduce .875 to a common fraction. Ans. The Ans. 314 7. Express 96.024 by an integer and a common fraction. Ans. 96135 8. Express 163.04 by an integer and a common fraction. 9. Express 1001.4375 by an integer and a common fraction. Ans. 1001 76 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. 916327át 278. To reduce a common fraction to a decimal. Ex. 1. Reduce s to a decimal. Ans. .375. OPERATION. Since we cannot divide the numer8 ) 3.0 (3 tenths. ator, 3, by 8, we reduce it to tenths 24 by annexing a cipher, and then di viding, we obtain 3 tenths and a re8) 60 ( 7 hundredths. mainder of 6 tenths. Reducing this 5 6 remainder to hundredths by annexing a cipher, and dividing, we obtain 7 8 ) 40 (5 thousandths. hundredths and a remainder of 4 40 hundredths; which being reduced to Ans. .375. thousandths by annexing a cipher, and then divided, gives a quotient of Or thus: 5 thousandths. The sum of the sev8) 3.0 0 0 eral quotients, .375, is the answer. To prove that .375 is equal to g, .37 5 Ans. we change it to the form of a com mon fraction, by writing its denominator, and reducing it to its lowest terms. Thus, .375 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. 3 7 5 EXAMPLES 2. Reduce { to a decimal. Ans. .625. 3. Reduce 1 to a decimal. 4. Changes to a decimal. Ans. .09375. 5. Change i} to a decimal. Ans. .076923+ 6. Reduce 19 to an equivalent decimal expression. 7. Reduce $ 315} to an equivalent decimal expression. Ans. $ 315.875. 8. Reduce $ 1163; to an equivalent decimal expression. Ans. 1163.75. NOTE. A decimal with a common fraction annexed constitutes what is called a complex decimal; as, .87, .314, and .18In such expressions, instead of the common fraction, its equivalent decimal, with the decimal point omitted, may be substituted. Thus, .425 .404. 9. Reduce .624 to a simple decimal. Ans. .625. 10. Reduce .3716 to a simple decimal. Ans. .370625. 11. Reduce $ 4.314 to a simple decimal expression. Ans. $ 4.3125. 12. Reduce $ 60.18to a simple decimal expression. Ans. $ 60.1875. 21 13. What decimal expression is equivalent to į of of 2.047 22 - 봉 2 + 0.371 OPERATION. 14. What decimal expression is equivalent to 2 8' + 3 of 4 of 4, 1.05 ? 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 £. Ans. .790625. We commence with the 3far., which 4 3.0 0 far. we reduce to hundredths by annexing d. 1 2 9.7 5 0 0 two ciphers; and then, to reduce these to the decimal of a penny, we divide by 2 0 1 5.8 1 2 5 0 s. 4far., since there will be į as many hun dredths of a penny as of a farthing, and .7 9 0 6 2 5 £. obtain .75d. Annexing this to the 9d., we divide by 12d., since there will be ih as many shillings as pence; and then, the 15s. and this quotient by 20s., since there will be to as many pounds as shillings, and obtain .790625£. for the answer. Hence the following 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. 60. 1£. 280d. ; 216 £. .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 60 24 or more, for THOUSANDTHS. Thus, if it be required to reduce, by inspection, 19s. 10d. 2far. to the decimal of a £.; half of 188. = 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 5.o£., 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 £. 186d.; EXAMPLES 2. Reduce 9s. to the fraction of a pound. Ans. .45. 3. Reduce 15cwt. 3qr. 141b. to the decimal of a ton. 4. Reduce 2qr. 21lb. 8oz. 12dr. to the decimal of a cwt. Ans. .71546875. 5. Reduce lqr. 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. lpt. 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. 1.fpt. to gallons and the decimal of a gallon. Ans. 156.6875. 15. Reduce to the decimal of a pound, 19s. 112d., 16s. 9fd., and 17s. 5 d., 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 £. ? Ans. 15s. 9d. 3far. There will be 20 times as many mil.7 90 6 2 5£. lionths of a shilling as of a pound ; there20 fore, we multiply the decimal, .790625, by 20, and reduce the improper fraction 1 5.8 1 2 5 0 Os. to a mixed number by, pointing off six 1 2 figures on the right, which is dividing by its denominator, 1000000. The figures 9.7 5 0 0 0 Od. on the left of the point are shillings, and 4 those on the right, the decimal of a shilling. 3.0 0 0 0 0 Ofar. The decimal" .312500 we multiply by 12, and, pointing off as before, obtain 9d., Ans. 15s. 9d. 3far. and a decimal of a penny. The decimal OPERATION. |