7. What is the quotient of 43 divided by 174? Ans. 8. What is the quotient of divided by 10? Ans. 14. 12. Divide the sum of 3, 4, 4, %, by the sum of 1, 1, 1, 13. Divide the sum of 4, %, 7, 4, 1, 4, H, H, 13, 14, by the sum of 1, 1, 1, 1, 7, 1, 1, to, t1, t2, 13. Ans. 4088833=47!!!!}. 845693= 84569 14. Divide of of of by of of 4 of 4. Ans. 13. 15. Divide the sum of 1, 11, 21, 31, by the sum of 1}, 2, 3ž. Ans. 48=1787. 16. Divide the sum of of, of, by the sum of of 4 of 7. Ans. H=14. 17. Divide of 14 of 4 by of 4 of 4. RECIPROCALS OF NUMBERS. 47. The reciprocal of a number is the result obtained by dividing 1 by the number. Thus, the reciprocals of 2, 3, 4, and 5, are 1,,, and . From this we discover that the reciprocal of an integer, or whole number, is equal to a vulgar fraction whose numerator is 1, and whose denominator is the given number. The reciprocal of is found by dividing 1 by, which (ART. 46,) is 1+3=1x5=5. In the same way we find the reciprocal of to be 4, and in general, the reciprocal of a vulgar fraction is the value of the fraction when inverted. NOTE. From this, we see that dividing by any number is in effect the same as multiplying by the reciprocal of that number. So that operations of division may be included under those of muitiplication. A practical application of this principle may be seen under Reduction of Denominate Fractions. (ART. 89.) EXAMPLES. 1. What are the reciprocals of 7, 8, 9, 10, 11? Ans.,,, To, tr. 2. What are the reciprocals of 18, 23, and 41 ? 6 Ans. 1, 2, . 3. What are the reciprocals of, 1, 4, t? Ans. 4, 4, 4, 4. 4. What are the reciprocals of 11, 21, 31? Ans. †,‡, 5. What are the reciprocals of 4 of 4, 4 of 7? 101 15. Reduce to a mixed number. 23. Reduce of 24. Reduce Ans. 3. Ans. 2557. Ans. . Ans. 20. Ans. 5. Ans. H. Ans. 17. Ans. . Ans. 1. of 29 to its simplest form. off of 3 to its simplest form. Ans. 1. of 3 of 21 of 3 to its simplest form. Ans. . 25. Reduce of of of 100 to its simplest form. Ans. 200. Ans. 1, 11, ਲੰ 26. Reduce,,, to equivalent fractions having a common denominator. 27. Reduce,,,,, to equivalent fractions having a common denominator. Ans. 30, 20, 5, 78, 78. 28. Reduce 3, 7, 4, , to equivalent fractions having a common denominator. Ans. 10, 28, 15, 26. 29. Reduce,,,, to equivalent fractions having a common denominator. 10 231 Ans. 15, s. 1156, 1166. 1155 1155: 1559 30. Reduce,,,, to equivalent fractions having a common denominator. Ans. 2003, 3575, 3085, 1835. 50059 4235 Ans. 1=177. Ans. 14-223. of the whole was cut 31. What is the sum of 1, 1, 1? 32. What is the sum of 4, 4, 5 ? 33. From a piece of cloth and off. What part of the whole was thus taken away? 37. A tree 150 feet high had broken off in a storm. What was the length broken off? Ans. 30 feet. 38. A and B together possess 1477 sheep, of which A and B. How many belong to each man? owns 39. A owns to B of his share. Ans. A's, 844. of a ship, valued at $15422; he sells What is the value of what A has left; also, what is the value of B's part? 40. A cotton mill is sold for $30000, of which A owns of the whole, B and C each own of of the whole. How many dollars does each one claim? A claims $6000. Ans. B claims $5000. LC claims $5000. 41. A and B have a melon, of which A owns 3, and B ; C offers them one shilling, to partake equally with them of the melon, which was agreed to. How must the shilling be divided between A and B? A must have of it. Ans. {B must have % of it. 42. A farmer had of his sheep in one field, † in a second field, and the residue, which was 779, in a third field. How many sheep had he in all ? Ans. 1230 sheep. 43. If I divide 616 dollars between A, B, C, and D, by giving A of the whole, B of the remainder, C & of what then remained, and D the balance, how much will each receive? A had 154 dollars. B 66 165 66 Ans. C 66 264 (6 D. 66 33 66 DECIMAL FRACTIONS. 49. A Decimal Fraction is that particular form of a Fraction, whose denominator consists of a unit, followed by one or more ciphers. Thus: o,, 180, 100, 180, 1000, 1000, &c., are Decimal Fractions. In practice, the denominators of Decimal Fractions are not written, but always understood. The above Decimal Fractions are usually written as follows: 0.1, 0.3, 0.04, 0.37, 0·08, 0·003, 0·0047, &c. The period, or decimal point, serves to separate the decimals from the whole numbers. The first figure on the right of the decimal point, is in the place of tenths; the second figure is in the place of hundredths; the third figure in the place of thousandths, and so on; the value of the units of the successive figures decreasing from the left towards the right, in a tenfold ratio, as in whole numbers. The following table will exhibit this. |