2. Reduce ⚫009, 0083, and '07954 to vulgar fractions. 3. Reduce 0185, 0046296, and 7621951 to vulgar fractions. 4. Reduce 446428571, and 254629, to vulgar fractions. Note. Circulates, which have the same number both of finite and circulating places, are called similar circulates. Two or more circulates may be made similar thus: "Point off by a comma on the right, as many figures from the left of each repetend as there are places in the longest finite part; then extend each of them as many places beyond that, as is denoted by the least common multiple of the numbers of places in the several repetends." EXAMPLE. Make 416, 63, and .296 similar. •416-41,666666 -63-63,636333 .296 29,629629 part of 2 places, we set off 2 figures from each repHere, because one of the repetends has a finite etend, and extend them 6 places farther, because 6 is the least common multiple of 1, 2, and 3, the numbers of places in the several repetends. ADDITION OF DECIMALS: 1st. To add finite decimals. RULE. Place the figures so that those of the same denomination (and consequently the decimal points,) may stand directly under each other; then add as in whole numbers, and place the decimal point in the sum directly under the other points. EXAMPLES. 1. Add together 5-5+61-75+ 12. Add together 23+9+4. ⚫0125. 23=23-75 Remark. The truth of the rule is obvious: thus, in the first example, if 3 ciphers are added to the first of the given numbers, and 2 ciphers to the 2d, the fractions will be reduced to a common denominator; their sum will then be equal to that of their numerators, placed over the common denominator: but it is plain, that the sum found by the rule is the same as it would be if these ciphers were added, and consequently it is truly found. EXERCISES. 1. Add together 17.5+182.75+4+19.85+008125+89 655. 2. Add together 89-8125 +271-05 +375 + 1279+01875 + 68.28945. 3. Add together 01825+17.5+.00375+199.25+144+310-0125.. 4. Add together 21+19+43+15+833+45%. 5. Reduce to decimals, and then add together £21: 10+£8:17:6 +£4:18:9+£3:3:6+£9:18: 7+£1 : 16:24. 2d. To add single repetends. RULE. Extend the repeating figures one place beyond the finite ones, and carry at 9 when you add the right hand column. Remark. As repeating figures signify 9th parts, the reason of the rule is obvious. EXERCISES. 1. Add together 83+7·416+·31855+6·25+4·38+29·627. 2. Reduce and add together £44: 7:6+£9: 15: 10+£6: 8: 8+ £12:19: 7+£10:0:04 +£9:0:24+£13: 9:51. 3. Reduce and add together £110: 11: 33+£97 : 11 : 84+ £45:13: 114+£912: 13:2+£35: 18:1}+£47:3: 111+ £58: 19: 1. 4. Reduce and add together 12+3+41,5+85+7+69+5%. 3d. To add compound repetends. RULE. Extend the repeating figures till they become similar, and when you add the right hand column, include the carriage that would have arisen, if the repetends had been extended further. Note. Circulates may also be added thus: "Take the sum of the repetends on a separate piece of paper, and divide it by a number consisting of as many 9s as there are repeating figures, the remainder will be the sum of the repetends, and he quotient must be carried to the next column on the left." EXERCISES. 1. Add together 9·45+5·3 + 13·83 + 1·76235 + 16·42135 + 157-025641+19-142857. 2. Reduce and add together, 2cwt. 3qrs. 7 lbs.+9cwt. 2qrs. 16lbs. +2qrs. 14lbs.+3 cwt. 1qr. 8lbs.+2cwt. 16 lbs.+5 cwt. 2qrs. 7 lbs.+16 lbs. 3. Reduce and add together 94+58,35, +152} } + 143 + 45§ + 97+76 SUBTRACTION OF DECIMALS. 1st. To subtract finite decimals. RULE. Place the figures so that those of the same denomination may stand under each other; then subtract as in whole numbers, and place the decimal point in the remainder directly under the other points. 1. From 5.53125 take 1.25. 2. From 213.5 take 1.8125. 3. From 45 take ⚫0045. 4. From 31. take ⚫875. 5. From 173 take 99 2d. To subtract repetends. RULE. Extend the repeating figures as in Addition, and if the repetends of the subtrahend exceed that of the minuend; suppose 1 added to the right hand figure of the former before you subtract. MULTIPLICATION OF DECIMALS. 1st. To multiply finite decimals. RULE. Place the factors under each other, and multiply as in whole numbers: then point off from the right of the product as many figures for decimals, as there are decimal places in both factors; when the product has not so many places, supply the defect by writ ing ciphers before it. Remark. The truth of the rule is evident; for the process it requires is to multiply the numerators of the given fractions together, and then apply to the product a decimal denominator, equal to the product of the given denominators; the fraction which results must therefore be the product sought. Thus, in the 1st example, 12375 × 188 1546875 15-46875; and in the 2d example, 1000X1800-18003125. 1000 = 1. Mult. 8.25 by 4.5. = EXERCISES: 6. Mult. 5-425 by 2.125. 9. Mult. 4.3125 by 100, by 1000, and by 10,000 separately. 2d. To multiply when the multiplicand is a repetend. RULE. When the multiplicand is a single repetend, carry at 9 when you multiply the repeating figure; when the multiplicand is a circulate, to the product of the right hand figure of each line, add the carriage that would have arisen, had the repetend been extended further; and make the repetends of the several lines of products similar before you add them. Note. When the product repeats, its repetend will be similar to that of the multiplicand. 2. When there are ciphers on the right of the multiplicand, multiply as if there were none; and when the work is done, extend the repeating figures in the product as many places as there were ciphers passed by. EXAMPLES. 1. Mult. 879-83 by •721 2. Mult. 586-1635 by 827. 3d. To multiply when the multiplier is a repetend. RULE. Reduce the multiplier to a vulgar fraction; then multiply by the numerator, and divide by the denominator. Note. When the divisor consists of three or more 9s, it is best to divide as in Rule III. page 20. EXAMPLES. 1. Mult. 157.525 by 46 2. Mult. 47-57185 by 9108 •46=·4§=·4·3=1 157.525 9108-9 .9108 •4% 630100 105016 73.5116 Ans. 9099 42814666 428146666 42814666666 9990)432856-28 Or thus. 157·525×775=73.5116. 43285 43 43.328956 Ans. Remark. When both factors are circulates, the number of decimal places in the product is often great, but it cannot exceed the product of the denominator of the multiplier into the number of places in the circle of the multiplicand. In practice, tedious circles are limited to as many places as will make the calculation sufficiently exact; as it would be idle to carry them further. EXERCISES. 1. Mult. 92.25 by 3 15 Mult. 23.148 by 3.90248. |