II. TO CONVERT A terminate DECIMAL INTO A VULGAR FRACTION. RULE.- Write the given decimal as the numerator, and for the denominator, write 1 and as many nothings as there are figures in the decimal : then reduce the fraction thus obtained to its lowest terms. Example.—Reduce •75 to a vulgar fraction. 2 = Answer. Exercises.—-Reduce the following to vulgar fractions1. •5 . . . Ans. } | 3. •25 . . Ans. $ 2. .04 . . 4. •78 . . 38 Note 1.--A pure recurring decimal is converted into a vulgar fraction by writing as many nines for the denominator as there are figures in the decimal, thus ·ż is written ģ; •si ass=ii: NOTE 2.-A mixed recurring decimal is converted into a vulgar fraction as follows: Subtract the non-recurring figures from the decimal, and write the remainder for the numerator : then for the denominator write as many nines as there are recurring figures, and annex to them as many nothings as there are non-recurring figures ; the resulting fraction may then be reduced to its lowest terms. Example.-Convert 7236 to a vulgar fraction. Here we deduct the non-recurring figures, 72, from the decimal, leaving 7236 less 72 = 7164 _ 199 7164 for the numerator, and then write 9900 275 two nines and two nothings for the de nominator : the resulting fraction is then reduced to its lowest terms. III. TO CONVERT A GIVEN SUM, AS 2s. 6d., TO THE DECIMAL OF , A HIGHER DENOMINATION. RULE.—Convert the given sum, when compound, to its lowest denomination; convert also one of the specified higher denomination to the same denomination as the other; annex as many nothings to the former as will make it greater than the latter; then divide the one by the other, continuing to annex nothings and to divide till there is no remainder, or as far as the division is wished to be carried. There must be as many decimals in the answer as there have been ciphers annexed. Example.-Reduce 2s. 6d. to the decimal of a pound. Here 2s. 6d. is reduced to its lowest 2s. 6d. denomination, pence = 30, and one of the specified higher denomination, pounds, is 240 ) 30000 also reduced to pence=240; nothings are then annexed to 30, and the dividend £•125 Ans. divided by 240. As three nothings have been annexed, there are three decimals in the answer. IV. TO FIND THE VALUE OF A DECIMAL OF A GIVEN DENO MINATION. RULE.-Reckon the decimal as so many of the given denomination; then divide it, as in Compound Division, by 10, if it consist of one figure; by 100, if it consist of two figures; and so on; using always as many nothings as there are figures in the decimal. Example. Find the value of £•375. £•375 Here, £•375 is reckoned as £375, and as there 20 are three figures in the decimal, we divide by 1000, according to Compound Division, Rule III., page 52 ; there being no pounds in the answer, we reduce £375 to shillings, and point off 3 6,000 figures, leaving 7s.; then reducing the figures pointed off to pence, we again point off 3 figures, Ans. 7s. 6d. leaving 6d. 7,500 12 NOTE.THE DECIMALS OF A POUND may in practice be conveniently valued by the following Rule, three decimal places being taken. Reckon the double of the first decimal as so many shillings, and the second and third decimals as so many farthings_less 1 farthing for every 25: thus—to find the value of £:364, reckon 3 as 6s, and 64 as 64 farthings, less 2 (for twice 25), making 62 farthings or ls. 3 d., and the answer is 7s. 31d. This rule will give the answer nearly correct: it will never be more than įd. too much or too little. Exercises. 1. £.182 = 3s. 7 d. 3. £•825 = 16s. 6d. 2. £.375 = 7s. 6. 4. £.924 = 18s. 5ed. ADDITION OF DECIMALS. RULE.--Write down the numbers in such a way that the points shall be directly under each other; thus having units under units, tens under tens, &c., in integers ; and in decimals, tenths under tenths, hundredths under hundredths, and so on; then proceed as in Simple Addition : the decimal point in the answer is placed directly below the other points. Example.-234.678 39.76 Exercises. 1. Add 637.4, 295•76, 4586.314, . . Ans. 5519.474 2. 9863.5, 275.146, 3912.78650, . 1 14051.4325 3. 7.9654, 10.12450, 46.754361, . . . 64:844261 4. 298.65, 475.672, 54.89008, 829.21208 (19.023, 107854, 736.93072, 15.391,1 . " 7.08365, •713926, 8327.591, •00086, 9101:519556 S 31.01, 162-718, •037, 8:6195, " 116.310, 38279, •00615, 27.382,) , 246.46544 200 SUBTRACTION OF DECIMALS. RULE.— Write down the numbers so that the points may be under one another, as in Addition; then proceed as in Simple Subtraction. When there are not as many figures in the upper as in the under line, nothings are supposed to be annexed to the former. Example.--From 643.157 take 29•76231 613.39469 Answer. Exercises. 1. From 9.267 take 6.7203, . . . Ans. 2.5467 14.796 12.605, . 2:191 19.876 3042361, . ID 16.833639 17.96432 ! 12:3745, .. 5.58982 11316847, . 7.83153 6. 316.281 30:379624, . , 285.901376 . riach i no es MULTIPLICATION OF DECIMALS. Rule.-Write down the multiplicand and multiplier without attending to the points, and multiply as in Simple Multiplication; then point off from the product as many decimals as are contained in both quantities : if the product does not contain as many, prefix nothings to make up the required number. Examples.-Multiply 3.061 by 2.5, and •2312 by .021. 2312 021 2312 in the two quantities, and only five 4624 in the product, two nothings must be 7.6525 Ans. prefixed to make up the number. .0048552 Ans. Note.-To MULTIPLY by 10, 100, or 1 with any other number of nothings annexed, it is only necessary to remove the decimal point as many places to the right, as there are nothings in the multiplier, thus—46°78 multiplied by 10, becomes 467.8. Exercises. 46°78 by 2:3, . . Ans. 107.594 321•76' 5.42, 1743.9392 45.021 1 .023, 1.035483 4. 1•3215 1.0051, ·00673965 97.236 1 10, 972:36 154.321, 100, 15432.1 274:93857 1 .0283, 7.780761531 3:1415967 3.795 11.9223594765 licorico 21 DIVISION OF DECIMALS. RULE.—Divide as in Simple Division, without attending to the points : then point off as many decimals in the answer, as the dividend contains more than the divisor. If the quotient has not as many figures as will allow of this, prefix the required nothings to make up the number. When the dividend has not as many decimals as the divisor, before dividing, annex as many nothings to the dividend as will make the decimals in both equal. When there is a remainder after dividing, the division may be - carried further by annexing nothings to the dividend, which, of course, must be taken into account in pointing off the decimals in the answer, Example 1.-Divide 3:36 by 2.1. 2:1 )3.36( 1:6 Here one decimal is pointed off in the 126 answer, as the dividend contains one deci mal more than the divisor. 126 Example 2.- Divide 3:36 by •105. -105 )3•360( 32 Before dividing, a nothing is here annexed 315 to the dividend, to make the decimals in the dividend and divisor equaland being 210 thus equal, there are no decimals in the 210 answer. Example 3.-Divide •336 by 21. 21: ).336( .016 Here the quotient is 16, but as the divi21 dend has three decimals and the divisor none, 126 the answer ought to have three decimals : a nothing, therefore, requires to be prefixed 126 to the quotient, to make up the number. NOTE.—To Divide by 10, 100, or 1 with any other number of nothings annexed, it is only necessary to remove the decimal point as many places to the left as there are nothings in the divisor; thus-1245 divided by 100, becomes 1.245. Exercises. 1. Divide 231•0 by 4.2, . . . . Ans. 55. 2. 1 36.21 21:3, . . . 1.7 3. 7.424, 32, . . . . . 23.2 124.5 , •15, 830 1 497.235 m 49.7235 1284127 100, 12.84127 1 16.7235 , 98.7629, •1693297 on 71.237 r.069184, . . . 1029 6857 iais iconic 10, SEE EXERCISES IN DECIMAL MONEY, APPENDIX, page 139. |