OPERATION CONTRACTIONS IN DIVISION. 95. Contractions in Division are abbreviated forms of dividing. CASX I. 96. When the divisor is a composite number. 1. Divide 2952 by 24, using the factors 4 and 6. SOLUTION 1st.–To multiply by 24 we may multiply by 6, and then multiply that product by 4; hence, to 4)2952 divide by 24 we may divide by 4, and then divide that quotient by 6. Dividing by 4 we have 738, and divid 6)738 ing 738 by 6 we have 123; hence, etc. 123 SOLUTION 2D.-Since 24 times a number equals 6 times 4 times the number, te of the number equals t of 1 of the number; 1 of 2952 is 738, and } of 738 is 123; hence, etc. Rule.-Divide the dividend by one factor of the divisor, the quotient by another factor, and thus continue for all the factors used ; the last quotient will be the quotient required. WBITTEN EXERCISES. Divide the following, using the factors: 2. 570 by 15. (15=3 x 5) 3. 492 by 12. (12=4x3) 4. 594 by 18. (18=3x6) 5. 2480 by 20. (20=425) 6. 4494 by 14. (142x) 7. 10950 by 30. (30=56) 8. 7875 by 35. 9. 12560 by 40. 10. 22824 by 72. 11. 47412 by 108. 12. 64440 by 120. 18. 54576 by 144. Ans. 38. Ans. 41. Ans. 33. Ans. 124 Ans. 321. Ans. 365, Ans. 225. Ans. 314. Ans. 317. Ans. 439. Ans. 537. Ans. 379. TO FIND TIE TRUE REMAINDER, 97. The True Remainder in successive division, it is evident, is not the last remainder, nor the sum of all the remainders; it is necessary, therefore, to explain the method of finding the true remainder. OPERATION. 1. Divide 791 by 24, using the factors 2, 3, and 4. SOLUTION.-Dividing by 2 we find that 191 equals 395 twos and 1 remaining ; di 2791 viding 395 twos by 3, we find 395 twos equals 131 sizes and 2 tron, or 4, remaining; di 3)395 1 viding by 4, we find that 131 sixes consists 4)131, 2 troos = 4 of 32 twenty-fours and 3 sizes, or 18, remain 32, 3 sizes = 18 ing. Hence the true remainder is 18+4+ 1, which is 23. Hence, to find the correct True remainder, 23. remainder we have the following Rule.--- Multiply each remainder by all the divisors pre ceding the one which obtained it, and take the sum of the products and the remainder arising from the first division Divide the following and find the true remainder: 2. 582 by 15. Rem. 12. 3. 503 by 12. Rem. 11. 4. 2497 by 20. Rem. 17. 5. 4507 by 14. Rem. 13. 6. 3717 by 30. 2, 3, 5. Rem. 27. 7. 13853 by 105. 3, 5, 7. Rem. 98. 8. 41837 by 180. 4, 5, 9. Rem. 77. 9. 47117 by 308. 4, 7, 11. Rem. 301, 10. 96711 by 310. 2, 5, 31. Rem. 301. 11. 87831 by 720. 2, 3, 4, 5, 6. Rem. 711. CASK II. OPERATION. 98. When there are ciphers at the right of the divisor. 1. Divide 8254 by 600. SOLUTION.-6 hundreds are contained in 82 hundreds 13 times, and 400 remaining; 600 is not contained in 6100)8254 64, hence the entire remainder is 400+54, or 454. From this solution we may derive the following 13-454 Rule.-I. Cut off the ciphers at the right of the divisor and as many terms at the right of the dividend. II. Divide the remaining part of the dividend by the remaining part of the divisor. III. Prefix the remainder to the part of the dividend cut off, and the result will be the true remainder. NOTES.--1. When the divisor is a unit of any order with ciphers, the romainder will be the figures cut off at the right, and the quotient the figures at the left. 8. When the part of the divisor at the left of the daughts is greater than 12, divide by long division. 2. Divide 876 by 50. Ans. 17; Rem. 26. 3. Divide 953 by 400. Ans. 2; Rem. 153. 4. Divide 1733 by 500. Ans. 3; Rem. 233. 5. Divide 2765 by 700. Ans. 3; Rem. 665. 6. Divide 7859 by 800. Ans. 9; Rem. 659. 7. Divide 9763 by 900. Ans. 10; Rem. 763. 8. Divide 14873 by 1900. Ans. 7; Rem. 1573. 9. Divide 25075 by 2300. dns. 10; Rem, 2075. 10. Divide 187654 by 14700. Rem. 11254, 11. Divide 269856 by 237000. Rem. 32856. 12. Divide 5767220 by 4730000. Rem. 1037220. EXERCISE UPON THE PARENTHESIS. 99. The Parenthesis (), denotes that the quantities included are to be subjected to the same operation; thus, (8+6-4)3 denotes that the value of 8+6—4 which is 10, is to be multiplied by 3. The vinculum, thus 8+6–4X3, is often used in place of the parenthesis. 1. What is the value of (12+9—7)X5? SOLUTION.—12+9 equals 21, and 21 minus 7 equals 14, and 14 mul. tiplied by 5 equals 70. Therefore, etc. Required the value 2. Of (25 +23-18) x 7. Ans. 210. 3. Of (46 +97-82) x 9. Ans. 549. 4. Of (98-75+87) x 14. Ans. 1540. 5. Of (89+96-47)+6. Ans. 23. 6. Of (145-110+117)+8. Ans. 19. 7. Of (396-128+483) x 32. Ans. 24032. 8. Of (869+990-1120)=-45. Ans. 16. 9. Of (320-98) x (860--145). Ans, 158730. 10. Of (689-327+986-397) x 428. Ans. 407028. 11. Of (729+487-244)+(247-210+71). Ans. 9. 12. Of (3014-2601) (2477-1325)+59. Ans. 8064. NOTE.-In a series of numbers connected with symbols, the sign X donotes the closest connection, the sign + next, thus, 12+8+2-BX? =-6; also, 16+4X2=2, rather than 8. WRITTEN EXERCISES. ON THE NUR FUNDAMENTAL RULES. 1. The minuend is 4160, and the subtrabend is 3425; wbat is the remainder ? Ans. 735. 2. The minuend is 9164 and the remainder is 3426 ; what is the subtrahend? Ans. 5738. 8. The subtrahend is 3872 and the remainder 4648; what is the minuend? Ans. 8520. 4. The multiplicand is 745 and the multiplier 456; what is the product ? Ans. 339720. 5. The multiplicand is 2463 and the prodrct 85466'; what is the multiplier ? A18. 347. 6. The product is 881919 and the multiplier 981 ; what is the multiplicand ? Ans. 899. 7. The dividend is 518077 and the divisor 763; what is tbe quotient? Ans. 679. 8. The dividend is 801222 and the quotient 3257 ; what is the divisor ? Ans. 246. 9. The divisor is 587 and the quotient 8723; what is the dividend ? Ans. 5120401. 10. The dividend is 72987 and divisor 45; required the quotient and remainder. Ans. 1621 ; 42. 11. The dividend is 7972, the quotient is 274, and remainder 26; what is the divisor ? Ans. 29. 12. The divisor is 26, the quotient 372, and renainder 23; what is the dividend ? Ans. 9695. 13. Thomas read 789 pages of history in a week, which lacks 324 of being as many as Walton read; how many did both read? Ans. 1902 pages.. 14. A freight car ran 365 miles one week, and 3 times as far, lacking 246 miles, the next week; how far did it run the second week ? Ans. 849 miles. 16. A sold 8318 bushels of wheat, then bought 2514 bushels, and then had 3146 bushels ; how many bushels had be at first? Ans. 8950 busbels. 16. My barn cost $3156; my house cost 3 times as mucb as my barn, and my farm cost as much as both ; what was the cost of all ? Ans. $25,248. 17. The value of 5 horses and 7 mules is $2436; now if the value of each mule is $208, what is the value of each horse ? Ans. $196. 18. A man left $2535 each to his four children, but one of them dying, the three remaining children divided the money; how much did each receive ? Ans. $3380. 19. Mr. Smith left $6264 to each of three sons and $7240 to each of two daughters, but one daughter dying, her share was equally divided among the remaining children; what did each receive ? Ans. Son, $8074; daughter, $9050. 20. The income of a man who “struck oil” was $480 a day ; how many teachers would this employ at a salary of $438 a year? Ans. 400. 21. A stock dealer bought 325 cows at $28 each, and sold 124 of them at cost; how much must be receive a head for the remainder to gain $804 ? Ans. $32. 22. Mr. Galto'ı buys a farm of 110 acres at $75 an acre, $2200 to be på d down and the remainder in five yearly installments; what must be pay each year? Ans. $1210. 23. A farmer raised 765 bushels of oats, of which he kept 65 bushels for seed, and after retaining enough for the use of bis borses till next harvest, allowing 60 bushels to each horse, sold the balance at 85 cents a bushel, and received $442 ; bow many horses had he ? Ans. 3 horses. 24. Mr. Milman bequeathed $6500 to each of two sons, to a third son $1000, $5000 to each of 3 daughters, and the balance of his estate, amounting to $25,000, to several benevolent institutions; the will, however, being set aside, the property was divided equally among his children; what was the share of each ? Ans. $9000. 25. If a soldier enlisting in the late war for 3 years, received a bounty of $930; then served one year as a private, at $13 a month, 6 months as a corporal, at $14 a month, and 18 months as a sergeant at $17 a month; what was the whole amount of his pay and his average pay per month? Ans. $41 a month |