RULE. If the repetend commences at tenths, consider the repetend as the numerator, and an equal number of 9's as the denominator. If there are integral figures in the repetend, annex as many zeroes to the numerator. If the decimal has a finite part, from the decimal, considered as a whole number, subtract the finite part for a numerator. For a denominator, write as many 9's as there are figures in the repetend, annexing as many zeroes as there are finite figures. 1. Reduce .018 to a fraction, and reduce the fraction to its lowest terms. 2. Reduce each of the following numbers to a fraction : 2.7; 89.1;.004; .0513; 6.649. 3. Reduce each of the following numbers to a fraction: .16394 ; 1.852906; 245.9; 06843. Addition, Subtraction, Multiplication and Division of Infinite Decimals can usually be performed with sufficient accuracy, by extending the decimals to six or eight figures each. In Multiplication and Division, if an exact result is required, we must first change the decimals to fractions. 1. Add 96.024, 37.81, 495,576, and 13.3. RULE FOR DIVIDING BY 9's. When the divisor consists of any number of 9's, increase it by 1, for a new divisor. Divide the dividend by this new divisor. By the same divisor, divide the integers of the quotient, and proceed in a similar manner, until a quotient is obtained less than the divisor. Add all the quotients together, observing the number of units carried from decimals to integers. Add this carriage to the right hand decimal figure, and the integers will represent the quotient, and the decimals the remainder. When all but the units' figure of the divisor are 9's, increase the divisor by the difference between the units figure and 10, and divide as above directed, multiplying each quotient after the first, by the number added to the divisor. Multiply the number carried from decimals to integers, by the number added to the divisor, and add the product to the decimals for the true remainder. If this increased remainder exceeds the divisor, increase the quotient by 1, and subtract the divisor from the remainder for the true remainder. EXAMPLE FOR THE BOARD. 8905.473 Divide 8905473 by 999. The divisor 8.905 increased by 1, is 1000. Dividing hy the 8 rule, and adding the quotients, we obtain 8914.386. There being 1 unit to carry from 8914.386 decimals, we add 1 to the right hand deci1 mal figure, and find the quotient is 8914. 8914.387 quotient. 387. This rule is founded on the decimal value of goo, and it will be easily seen that the process is nearly the same as in the multiplication by .boiooit. In dividing the numerators of fractions obtained by the multiplication of circulating decimals, the rule will often be of use. Divide 1549638144 by 9991. The 154963.8144 divisor increased by 9, is 10000. Di139.4667 viding first by this number, we multi1251 ply the integers of the first quotient by 9, writing the first figure of the product 155103.4062 under the right hand decimal figure, 1X 9=9 which is equivalent to multiplying by 9, and dividing by 10000. We mul.4071 remainder. tiply the integers of this second number, and write them in the same manner, and add the several numbers together. There being 1 unit to carry from decimals, we multiply it by 9, and add the product to 4062, which gives 4071 for the true remainder. 1. Divide 870468 by 999; by 99. 3. Divide 7963142 by 999000. Divide first by 999, then divide the quotient by 1000. 4. Divide 1046.84 by 993 ; by 9991. 5. Divide 87364001 by 99999. CHAPTER VII. COMPOUND NUMBERS. When several denominations of the same kind (as pounds, shillings and pence; miles, feet and inches; gallons, quarts and pints) are embraced in one sum, they are called COMPOUND NUMBERS. The operations upon them, are performed by regarding each denomination as a fraction of the next higher. REDUCTION OF COMPOUND NUMBERS. EXAMPLE FOR THE BOARD. in. yd. ft. How many inches in 27 r. 4 yd. 27 6'4" 2 6 11 2 ft. 11 in. ? As there are 5.5 yards 5.5 in 1 rod, in 27 rods there will be 148.5 yds. In 27 r. 4 yd. there will 135 then be 152.5 yd. As 3 ft. make 1 135 yd., in 152.5 yd. there will be 457.5 ft. Adding the 2 ft. we obtain 459.5 ft. As 12 in. make 1 ft., in 459.5 ft. there are 5514 in. Adding the 11 in. we have 5525 in. for the answer. 3 Hence, to reduce the higher denomi nations of a compound number to 457.5 ft. their value in a lower denomination, we have the following rule. 148.5 yd. 4 yd. 152.5 yd. 2 ft. 459,5 ft. 12 5514.0 in. 11 in. 5525 in. RULE. 66 2 ft. 28 r. Commence with the highest denomination, and multiply each denomination by the number required of the next lower to compose it, adding to the product the number (if any) already in that denomination. Thus proceed until you have reached the lowest denomination sought. 12)5723 How many r. yd. ft. and in. in 5723 3)476 ft.“ 11 in. 476 ft.=476 yd.=158 yd. 2 ft. 158 yd.= inches ? 5723 in.=573 ft.=476 ft. 11 in. 5.5)158 yd. 158.0 r.=28°r. 4 yd. The answer is, therefore, 28 r. 4 yd. 2 ft. 11 in. Hence, “ 4 yd. to reduce the lower denominations of a compound number to their value in higher denominations, we have the following RULE. Divide each denomination by the number required of it to make 1 of the next higher, and give to each remainder the name of the dividend from which it is derived. 1. In 41 chal. 19 bu, of coal, how many pecks? 2. How many cwt. &c., in 1870953 gr. .? 3. How many th, &c., in 24753 gr. Troy? 4. Reduce 1951 113 33 to grains. 5. Reduce 69741895 inches to miles, &c. 6. Reduce jo cwt. to gr. lb. &c.* 7. Reduce £9.481 to £ s. &c. 8. Reduce 7964 gills to gallons, &c. 9. Reduce 43 Y, to days, &c. 10. Reduce 51 { Cong. to Cong. O., &c. 11. Reduce 8753 in. to yd. qr., &c. 12. How many acres in a piece of land that is 15r. 4yd. long, and 14 r. 2 yd. 2 ft. wide ? 13. How many cords in a pile of wood 21.5 ft. long, 8.3ft. wide, and 9.7 ft. high? 14. Reduce 4. Y. to seconds. * The reduction of fractions and decimals is performed in the same way as reduction of whole numbers, by multiplying or dividing, as the question may require. 15. How many tons of hewn timber in 19 sticks, each stick being 18.5 ft. long, 1.3ft. wide, and 1.1 ft. thick? 16. How many drops of distilled water would fill a cistern that is 6 ft. din. long, 5 ft. 3in. wide, and 4 ft. 2in. deep? [First find the number of cubic inches in the cistern, then divide by 231 (because 231 cubic in, make a Cong.), and multiply by the number of drops in a Cong.] 17. What is the value of £197 7s.6d. in Federal Money, estimating the penny at 2 cents ? 18. What is the value of 6971 roubles 25 copecks, at of a cent per copeck ? 19. Reduce 1 of a mile to fur., r. &c. 20. Reduce 10 pounds 17 shillings 10 groats Flemish, to dollars and cents, at a cent a groat. 21. Reduce 1728431||/ to ft., primes, &c. 22. The specific gravity* of gold is 19.258. Then, what is the weight of a mass of gold that is 6 inches long, 5 inches wide, and 3 inches thick ? TO REDUCE LOWER DENOMINATIONS TO THE FRACTION OF A HIGHER. What part of a £ is 2s. 7d. 3qr.? 2s. 7d. 3qr.=127qr.='9'd=127 of i's.=137 of 'of £ RULE. First reduce the given sum to the lowest denomination mentioned. With this result make a fraction of the next higher denomination, then a compound fraction of the next higher, and so proceed until you reach the denomination required. 1. Reduce 9s. 3d. 2qr. to the fraction of a £. 2. Reduce 3qt. 1pt. 2gi. to the fraction of a gallon. 3. Reduce 13h. 11 min. 11 sec. to the fraction of a day. 4. Reduce 4yd. 10in. to the fraction of a rod. 5. Reduce 70. 6f 3, to the fraction of a gallon. 6. Reduce 7s. 6d. to the fraction of a guinea. * The specific gravity of a body, is its weight compared with an equal bulk of water. Thus, gold being 19.258 times as heavy as water, its specific gravity is 19.258. |