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. 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 7 is written 3; 81 as 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. 7236 less 72 7164 199 9900 275 Here we deduct the non-recurring figures, 72, from the decimal, leaving 7164 for the numerator, and then write two nines and two nothings for the denominator: 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. 2s. 6d. 12 240)30000 £125 Ans. = Here 2s. 6d. is reduced to its lowest denomination, pence 30, and one of the specified higher denomination, pounds, is also reduced to pence = 240; nothings are then annexed to 30, and the dividend divided by 240. As three nothings have been annexed, there are three decimals in the answer. F Exercises. 1. Convert 12s. 6d., 5s. 4d., and 6s. 34d. to the decimal of £1, Ans. £625, £26, £314583 17s. 52d., 18s. 74d., and 13s. 6d. to the decimal of £1, Ans. £8739583, £•93125, £675 2. 3s. 104d., 19s. 9d., and 16s. 8d. to the decimal of £1, Ans. £1927083, £·9875, £·83 4d., 6d., and 8d. to the decimal of £1, Ans. £016, £025, £.03 2 qrs. 17 lbs. to the decimal of a cwt., Ans. 6517857142 cwt. 3 cwts. 3 qrs. 8 lbs. to the decimal of a ton, Ans. 1910714285 ton. NOTE. TO CONVERT SHILLINGS, PENCE, AND FARTHINGS into the decimals of a pound, the following is a convenient rule in practice: Reckon half the number of the shillings as the first decimal; thus, consider 12s. as 6 if the number of shillings is odd, as 13s, carry the odd 1s. to the pence, convert the pence, also the odd ls. into farthings, and include any farthings that are in the given sum; then reckon the farthings as the second and third decimals, adding 1 for every 24; thus, if the farthings come to 54, reckon the decimal as 56 (for twice 24). This rule gives the answer nearly correct; it will never be more than too much or too little. Example.-Convert 13s. 6d. into the decimals of £1, Ans. £.677 Here the half of 13s. is 6s. and 1 over: the 6 is placed as the first decimal; then the odd 1s. and the 6d. 1s. 6d. are 74 farthings, and adding 3 farthings (for thrice 24), make 77 farthings, which are placed as the second and third decimals. 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 7,500 12 6,000 Ans. 7s. 6d. Here, £375 is reckoned as £375, and as there 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 figures, leaving 7s.; then reducing the figures pointed off to pence, we again point off 3 figures, leaving 6d. Exercises. 3. 1. Find the value of 75s., 2. 5. 6. 7. 8. 1.475 tons, Ans. £6375; £•78125, "I £1.928125; £15625, " £1, 18s. 63d.; 3s. 1d. 1 ton 9 cwts. 2 qrs. " 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 1s. 3d., and the answer is 7s. 3d. This rule will give the answer nearly correct: it will never be more than d. too much or too little. 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. 80 SUBTRACTION OF DECIMAL S. 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 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. 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. 81 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 3.-Divide 336 by 21. 21.).336(⚫016 21 126 126 Here the quotient is 16, but as the dividend has three decimals and the divisor none, the answer ought to have three decimals: a nothing, therefore, requires to be prefixed 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-124 5 divided by 100, becomes 1.245. SEE EXERCISES IN DECIMAL MONEY, APPENDIX, page 139. |