1. In addition and subtraction we obtain this result most easily by using in our operation one or two more figures of the given decimals than are required in the result. Ex. 1.-Add together (correct to five places) the following •3026, 6-7294, 016, .4163729. -3026026 6.7294444 .0166666 -4163729 7.4650865. Ans. 7.46508. NOTE. - If our object is to obtain a decimal of five places which shall give the approximate sum of the given decimals we must write 7.46509 as the approximate sum; for 7 46509 is nearer to the true value of 70465086 &c. than 7.46508. The general rule is to increase by 1 the decimal figure at which we stop, when the next figure is 5 or above 5. Ex. 2.—Find the difference (correct to six places) of 3.0745 and 4.263, and express the approximate difference by a decimal of five places. 4.26326326 1.18871781 Hence the difference correct to six places is 1.188717, and the required approximate difference 1.18872. 2. In multiplication and division of circulating decimals it is generally preferable to reduce the given decimals to fractions, bring out the result in a fractional form, and afterwards reduce this to a decimal. Ex. VII. 1. Express as a decimal the sum of the following fractions:音,古言, r,音高 ,, 2. Reduce to fractions the following decimals : •35, 026, 16, 142857, 16, +4285714. 3. Find the value (correct to six places) of 237 + 3.816 – 6.0235 + 4.29 - 002 + 1-374. 2.2 x 4. Add together •62, .037, 2.476i, •8106, -7, •375. (4.6 x 428571 -36) (1 – 16). In the next six examples the dots are signs of multiplication, and you are required to give the values of the expressions correct to six places. 1 1 1 1 7. 1 i 1.2 1.2.3 1.2.3.4 1 1 1 1 3.33 5.36 7-37 1 + &c. 1.2.3.4.5 1 1 1 10. 1 82 84 g6 + + + + + &c. + + &c. 1 + + + + + &c. To find the Value of a Fraction of a Concrete Quantity. 28. RULE.—Multiply the concrete quantity by the numerator of the fraction, and divide the product by the denominator. 6 tons, 9 cwt. 2 qrs. 7 lbs. 7 7 = £. 8. Ex. 1.–Find the value of 1 of 2 tons, 3 cwt. 21 lbs. (2 tons, 3 cwt. 21 lbs.) x 3 of 2 tons, 3 cwt. 21 lbs. == 18 cwt. 2 qrs. 1 lb. Ans. Ex. 2.—Find the value of 33 of £12. 6s. 2 d. d. 3 times £12. 6s. 24d. (£12. 6s. 2 d.) × 3 = 36 18 63 9 2 14 81 9 Hence, adding, the value required = £39 13 34 Ex. 3.–Find the value of of 4 miles + f of 3 fur. + at of 8 poles. Mile. Fur. Poles. Yards. of 4 miles = 1 5 28 34 7 of 3 fur, 0 2 13 18 As of 8 poles 0 0 1 401 Hence, adding, the value required 2 0 3 441 NOTE.—The addition of the yards is thus effected : (34 + 1f + 441) yds. (8 + 8+56,44) yds. (8 + 1/f) yds. 93 i yds. 1 pole, (3} + 4f) yds. = 1 pole,(3 + 1*.*°) yds. 1 pole, 41yds. 12 miles 7 (4 x 3) miles 7 9 21 9 40 poles 21 Ex. VIII. Find the values of 1. of £l; of 1s. ; & of 12s.; of £3. 2. 1 of £5; 4 of a guinea ; 0 of 2s. 6d.; of a crown. 3. 34 of £1. 12s.; 24 of £3. 10s.; 78 of £3. 4s. 54. of 1 ton; } of 1 qr.; of 1 stone; of 9 lbs. 5. 34 mile; of 3 fur.; of 15 poles ; 34 of 2} of 12 yds. 8 2 5 6. leap year; lunar month; 201 of 10 h. 15' 12". 13 10 7. 12 lbs. 3 oz. 7 dwt. 5 grs. x 34; 10 oz. 4 gr. • 16}. 4. $ 1 of 1 8. 8 of 4 ac. 3 po. sq. mile + 1ff of 3 r. 20 sq. yds. 9. (5 of 3x + 116) of 35° 36' 25"; (276 + 1 of 13) of 30°. 10. 1 of 1 lb. Troy + 3 of 1 lb. Troy - zo of 1 lb. Avoirdupois (= 7000 grains). 54 of 3 guineas 11. of fof £1. 3s. 10d. ; $ of 1 ton + 4 of 3 qrs. - It of 7 cwt. 12. 27 of 15 h. 10' 13" -of 1 day, 12 h. 11' 127". To Reduce a given Quantity to the Fraction of any other given Quantity of the same kind. 29. RULE.—Reduce both the given quantities to the same denomination, and the fraction required will have the number of units in the first quantity as numerator, and that of the second quantity as denominator. Ex. 1.-Reduce 3s. 8d. to the fraction of £1. 3s. 8d. 44d, and £1 240d. Hence, the fraction required 46 *t. Or, better, thus : 3s. 8d. = 11 fourpences, and £1 = 60 fourpences. : Fraction required = 1 NOTE. - It is always best to keep the denominations to which the given quantities are reduced as high as possible. Ex. 2.--Reduce 4 of a moidore to the fraction of 21 guineas. 1 of a moidore (4 x 27)s., and 2} guineas = (21 x 21)s. Hence, the fraction required 21 x 21 ** 27 x 21 1 X27X2 5X21X7 Ex. 3.-Reduce 3 cwt. 24 qrs. to the fraction of 4 cwt. 2 qrs. 4 lbs. 3 cwt. 24 qrs. 14} qrs., and 4 cwt. 2 qrs. 4 lbs. = 4 cwto 184 qrs. 141 1 2 7 X 7 .:: Fraction required 7. 24 qrs. 187 ܐܐ 127 9 9 X 127 Ex. IX. Reduce— 1. ls. 8d. to the fraction of £1; 7d. to the fraction of 10s. 2. 2s. 4d. to the fraction of 10s. 8d.; ls. 7 d. to the fraction of 3s. 4 d. 3. 3 qrs. 15 lbs. to the fraction of 1 ton; 2 stones 10 oz. to the fraction of 3 cwt. 4. 3 lbs. avoirdupois to troy weight; 10 lbs. 3 oz. 4 dwt. troy to avoirdupois. 5. 3 quires, 10 sheets to the fraction of 2 reams, 3 quires; 3 ft. 81 in. to the fraction of 3 yards. 6. 30° 3' 12" to the fraction of a right angle ( = 90°); 57° 16' 21/" to the fraction of two right angles. What fraction is7. fac. of 3} ac.; 24 days of 17 weeks ? •125 + 1.875 '3.16 x 1.4 8. acres of 19 poles ; yds. of 3? m? 0140625 2.375 xi 3 72 l} x 114 9. rood + poles + yd.) of 3 acres ? 47 21 o 유 10. 219 pipes ? 11. What fraction of his original income has a person left after paying a tax of 4d. in the £? 12. A garden roller is 2 ft. 6 in. wide, and it is rolled at the rate of 1 mile in 20 minutes : find in what fraction of a day a man will roll } of an acre. ( 326 To Find the Value of a Decimal of a Concrete Quantity. 30. RULE.—Multiply the given decimal by the number of units in the concrete quantity when expressed in terms of one denomination, and the integral part of the result will be the number of units of this denomination. Then multiply the decimal part of this denomination by the number of units connecting it with the next lower and the integral part will be units of this latter denomination, and so on. |