OPERATION. Rem. It will be remembered, that to mul110) 3516 tiply by 10 we annex one cipher, which Quotient 35, 6 Rem. removes the figures one place to the left, and thus increases their value ten Or thus, 3 5 6. times. Now, it is obvious that if we reverse the process, and cut off the right-hand figure by a line, we remove the remaining figures one place to the right, and consequently diminish the value of each ten times, and thus divide the whole number by 10. The figures on the left of the line are the quotient, and the one on the right is the remainder, which may be written over the divisor, and annexed to the quotient. Hence the share of each man is 3576 dollars. EXAMPLES FOR PRACTICE. Quotient. 2. Divide 6892 by 10. 689 2 3. Divide 4375 by 100. 75 4. Divide 24815 by 1000. 815 5. Divide 987654321123 by 100000000. 54321123 Art. 58. When the divisor has ciphers on the right, and is not 10, 100, &c. Ex. 1. If I divide 5832 pounds of bread equally among 600 soldiers, what is each one's share ? Ans. 9433 pounds. The divisor, 600, may 1100) 58/32 be resolved into the fac tors 6 and 100. We first 6)58, 32, 1st Rem. divide by the factor 100, 9, 4, 2d Rem. by cutting off two figures at the right, and get 58 Or thus, 6100)58 3 2 for the quotient and 32 for a remainder. We , 4 3 2 then divide the quotient, 58, by the other factor, 6, and obtain 9 for the quotient and 4 for a remainder. The last remainder, 4, being multiplied by the divisor, 100, and 32, the first remainder, added, we obtain 432 for the true remainder (Art. 56). Hence each soldier receives 943pounds. Art. 59. From the preceding illustrations is deduced the general RULE. Cut off the ciphers from the divisor, and the same number of figures from the right of the dividend. Then divide the remaining figures of the dividend by the remaining figures of the divisor. OPERATION. QUESTIONS. Art. 57. How do you divide by 10? How does it appear that this divides the number by 10?— Art. 58. How do you divide by 600 in the example ? How does it appear that this divides the number? - Art. 59. What is the general rule ? NOTE. — When by the operation there is a last remainder, to it must be annexed the figures cut off from the dividend to form the true remainder. Should there be no last remainder, then the significant figures, if any, cut off from the dividend, will form the true remainder. EXAMPLES FOR PRACTICE. Quotients. Rem. 2. Divide 3594 by 80. 44 74 3. Divide 79872 by 240. 332 192 4. Divide 467153 by 700. 667 253 5. Divide 13112297 by 8900. 2597 6. Divide 71897654325 by 700000000. 497654325 7. Divide 3456789123456787 by 990000. 306787 8. Divide 967231731328000 by 1020000000. 411328000 9. Divide 33166405115000 by 1600000000. 5115000 10. Divide 18191618562300 by 1000000000. 618562300 11. Divide 4766666000000 by 55550000000. 44916000000 VI. QUESTIONS INVOLVING FRACTIONS. ART. 60. If a unit or individual thing is divided into parts, one of the parts is called a Fraction of the number or thing divided. Hence FRACTIONS are parts of whole numbers. ILLUSTRATIONS. - 1. When any number or thing is divided into two equal parts, one of the parts is called one half, and is written thus : Ž 2. When any number or thing is divided into three equal parts, one of the parts is called one third (3); two of the parts are called two thirds (?). 3. When any number or thing is divided into four equal parts, one of the parts is called one fourth (); three of the parts, three fourths (X). 4. When any number or thing is divided into five equal parts, one of the parts is called one fifth (1); two parts, two fifths (3); three parts, three fifths (*); and four parts, four fifths (%). QUESTIONS. Art. 60. What is a fraction? What is meant by one half of any number or thing? How is it written ? What is meant by one third, and how is it written ? What by one fourth, and how written ? What by one fifth, and how written ? What by four fifths, and how written? How do you find one half of any number? How one third ? How one fourth ? &c. How many halves make a whole one? How many thirds ? How many fourths ? How many fifths ? 5. When any number or thing is divided into six equal parts what is one of the parts called ? Two parts? Five parts ? 6. When a number or thing is divided into 7 equal parts, what is 1 part called ? 2 parts? 3 parts? 4 parts? 5 parts ? 6 parts? 7. When a number or thing is divided into 9 equal parts, what is 1 part called? 2 parts ? 4 parts ? 5 parts ? 7 parts ? 8 parts? 8. What is 1 half of 4? Of 8? Of 16? Of 20? Of 28? Of 32 ? 9. What is 1 third of 9? Of 12? Of 15? Of 27 ? Of 30? Of 36 ? Of 60 ? 10. What is 1 fourth of 8? Of 16? Of 20? Of 24? Of 40 ? Of 48? Of 100? 11. What is 1 fifth of 10? Of 25 ? Of 30? Of 35? Of 45 ? Of 50? Of 55 ? Of 65 ? 12. What is 1 sixth of 12 ? Of 18? Of 30? Of 42 ? Of 60? Of 72? Of 90 ? 13. How many fourths in 1 apple? 14. How many fourths in 2 apples ? In 3 apples? In 8 apples ? In 16 apples ? 15. How many fifths in 1 barrel of flour? In 3 barrels ? In 5 barrels? In 7 barrels ? In 9 barrels ? 16. How many sixths in 1 bushel of wheat? In 4 bushels ? In 7 bushels ? În 9 bushels ? In 12 bushels ? 17. James owns 3 fifths of a kite, and his brother Thomas the remainder. How many fifths does Thomas own? ILLUSTRATION. Since there are 5 fifths in the kite, if James owns 3 (fths, there will remain for Thomas 5 fifths (5) less 3 Ans. 2 fifths. 18. From a load of hay I sold 4 sevenths; how many sevenths remain ? 19. John Jones found a large sum of money ; he gave 5 eighths of it to the poor of the parish ; how much did he reserve for himself? 20. John Smith gave 2 ninths of his farm to his son, 3 ninths to his daughter, and the remainder to his wife ; how many ninths did his wife receive ? ILLUSTRATION. Since he gave 2 ninths (s) to his son, and 3 ninths (8) to his daughter, he gave them both f + g = $; and since there are 9 ninths ($) in the farm, he must have given his wife - ģ= $. fifths (3) = 2 fifths. Ans. 21. In a certain school § of the pupils study grammar, i study arithmetic, ti geography, and the remainder philosophy. What part of the school study philosophy ? 22. J. Dow spends of his time in reading, 4 in labor, and & in visiting. How large a portion of his time remains for eating and sleeping ? 23. If a yard of cloth cost $8, what cost of a yard? What cost of a yard? ILLUSTRATION. If 1 yard cost $8, I of a yard will cost of $8= $2; and if of a yard cost $2, will cost three times as much ; 3 times $2 = $6. Ans. $6. 24. If an acre of land cost $24, what will f of an acre cost ? What will cost ? 25. When 96 cents are paid for a bushel of rye, what cost 17 of a bushel ? 26. If of a barrel of flour cost 2 dollars, what cost of a barrel ? ILLUSTRATION. — If } cost 2 dollars, f will cost 4 times 2 dollars $8. Ans. $8. 27. If of an acre of land cost $24 dollars, what will į of an acre cost ? 28. If of a hogshead of molasses cost $11, what will a hogshead cost ? 29. If y of an acre of land cost $21, what cost f of an acre ? What cost an acre ? What cost 10 acres ? ILLUSTRATION. — If } cost $21, ß will cost of $21, and 7 of $21 is $3; and g will cost 8 times $3 = $24, and 10 acres will cost 10 times $24 - $240. Ans. $240. 30. If I of a hogshead of sugar cost $18, what costs 1 hogshead? What cost 4 hogsheads ? 31. If of a barrel of apples cost $1.50, what costs a barrel ? What cost 10 barrels ? 32. When $49 are paid for 11 of a ton of potash, what must be paid for 2 tons ? 33. How many half-barrels of flour are there in 2 and a half (24) barrels ? ILLUSTRATION. - Since 1 barrel contains 2 halves, 2 barrels will contain 2 times 2 = 4 halves, and the 1 half added makes 5 halves. 34. How many half-bushels in 41 bushels of oats? In 57 bushels ? In 74 bushels ? In 91 bushels ? Ans. 35. How many eighths of a dollar in 2f dollars ? In 48 dol. iars? In 75 dollars? In 97 dollars ? In 123 dollars ? 36. How many tenths of an ounce in 41% ounces? In 5% ounces? In 85 ounces? In 1025 ounces? 37. How many barrels of wine in 6 half () barrels ? ILLUSTRATION. Since it takes 2 halves to make one whole one, there will be as many whole barrels in 6 halves (1) as 2 is contained times in 6. 2 is contained in 6, 3 times. 48? Ans. 3 barrels. 38. How many firkins of butter in z firkins? In z firkins ? 39. How many whole numbers in 1? In ? In 25? 40. How many whole numbers in je? In š? In 32? In 41. If a skein of silk is worth 31 cents, what are 6 skeins worth ? ILLUSTRATION. — If 1 skein is worth 31 cents, 6 skeins areworth 6 times as much ; 6 times 31 are equal to 6 times 3 and 6 times t; 6 times 3 = 18; 6 times ; = 3; 18 + 3 21. Ans. 21 cents. 42. Bought one pair of boots for $61; what must I pay for 4 pairs ? For 8 pairs ? For 10 pairs ? For 12 pairs ? 43. Paid 124 cents for one pound of cloves; what will 6 pounds cost ? 10 pounds ? 12 pounds ? 44. If one pound of butter is worth 12 cents, what are 451 pounds worth? ILLUSTRATION. — If 1 pound is worth 12 cents, 41 pounds are worth 41 times as much ; 41 times 12 cents are equal to 4 times 12 and 1 of 12; 4 times 12 are 48, and 1 of 12 is 6; 48 cents and 6 cents are 54 cents. Ans. 54 cents. 45. When lard is sold for 9 cents per pound, what must be paid for 74 pounds ? For 84 pounds ? For 94 pounds? 46. Bought 1 pound of coffee at 16 cents; what will 53 pounds cost ? 31 pounds ? 51 pounds ? 6j pounds ? 47. If 1 yard of cloth is worth 20 cents, what is the value of 161 yards? 121 yards? 81 yards ? 114 yards ? 48. If 11 bushels of corn cost $1.20, what will 1 bushel cost ? ILLUSTRATION. — 11 bushels bushels. Now, if cost $1.20, į will cost of $1.20 = $0.40; and for a whole bushel will cost 2 times $0.40 = $0.80. Ans. $0.80. 49. If 24 pounds of coffee cost 60 cents, what will 1 pound cost ? |