CASE III. 182. To divide a fraction by a fraction. 1. Divide by SOLUTION.- divided by one equals 1, hence divided by equals 6 times 3, and divided by 5 sixths equals of € times 7, which is times, which equals 13, or 3. Therefore we have the following OPERATION. 1+1= Rule I.-Multiply the dividend by the denominator of the divisor, and divide by the numerator. Rule II-Invert the divisor and proceed as in multiplication of fractions. NOTE.-Reduce mixed numbers to simple fractions. When the divisor is a compound fraction, invert each term and multiply, cancelling when possible. 1. If a pound of brown sugar cost 7 cents, how many pounds can you buy for 25 cents? Ans. 31. SUGGESTION. The analysis gives 1549, which by cancelling and multiplying equals 3. 2. If a quart of vinegar cost 183 cents, how many quarts can be bought for 52 cents? Ans. 24. 3. If 7 yards of book muslin cost $537, how much will 5 yards cost? Ans. $4.10. 4. If 15 yards of ribbon cost 45 cents, what will 6 yards cost? Ans. 174 cts. NOTE. In many of the problems it will be best to express the operation and saneel common factors. 5. If 21 pounds of tea cost $183, what will 1 pound cost? Ans. $0.85. 6. If 22 barrels of sugar cost $308, what is the price per barrel? Ans. $137. 7. If 284 tons of Lehigh coal cost $183g, what is the price per ton? Ans. $6. 8. Divide of off by § of 14 of 2 of 51. Ans. 12. 9. Find the value of 175÷337. Ans. 1. Ans. 4. 10. Find the value of (1×) ÷ (44×33). 11. Find the value of (2-1), and the product divided by (63-53). Ans. 3. 12. Find the value of (+34) 93, and the product divided by (6-41.) Ans. 161. 13. Find the value of (772444)X(68-53) divided by (8+73-374). Ans. 24. 14. If an errand boy earn $77 in a week, how long will it require him to earn $201? Ans. 24 weeks. 15. A lady distributed $231 among the poor, giving $287 to each person; how many were there? Ans. 8 persons. 16. Mr. B divided $416 equally among five persons; how much did each receive? Ans. $831. 17. By his father's will, Henry shared $9600 equally with his five brothers; how much did he receive? Ans. $1600. 18. The product of two numbers, diminished by 112, is 127; if one number is 15, what is the other? Ans. 163. 19. A boy shared 102 apples with his companions, giving to each 6 apples; how many companions had he? Ans. 15. 20. A man divided 30 pounds of flour among the poor, giving to each 2 pounds; how many persons were there? Ans. 11 persons. 21. A seamstress bought a sewing-machine for $85.50, and paid $40 down; how much must she save each month, to pay for it in 7 months? Ans. $6.50. 22. A man's wages are $3 a day and his daily expenses are $13; how many days must he labor to enable him to buy a suit of clothes worth $46? Árs. 24 days. 23. Mr. Landis gained $228 on the sale of 67 acres of land; how much more would he have gained on each acre, if he had gained $382 on the whole ? Ans. $22.40. REDUCTION OF COMPLEX FRACTIONS. 183. The Reduction of Complex Fractions is the process of changing them to simple fractions. NOTE.-A complex fraction is not really a fraction, according to the definition of a fraction. It is rather a complex fractional expression of one fraction divided by another. 1. Reduce to a simple fraction. Rule I.-Multiply the numerator of the complex fraction by its denominator inverted. Rule II.-Multiply both terms of the complex fraction by the least common multiple of the denominators. RELATION OF NUMBERS. 184. The Relation of Numbers is their relative value as compared with one another. NOTE. This subject is equivalent to Ratio, but is presented here as affording an excellent illustration of the analysis of numbers. The treat ment of the subject under Ratio is demonstrative rather than analytic. CASE L 185. To find the relation of an integer to an integer. 1. 27 is how many times 6, or what is the relation of 27 50 6? SOLUTION.-One is of 6, and if 1 is of 6, 27 is 27 times one-sixth of 6, which is 27, or 2, or 4 times 6. Therefore 27 is 4 times 6. Hence we have the following Rule. Divide the number which you compare by the number with which it is compared. NOTE. The rule is the same for each case, and need not be repeated. Ans. 13. 4. 76 to 24? 5. 84 to 91 ? 10. If 16 apples at the same rate? 6 Ans. 1.9. 1899 to 2743? cost 13 cents, what will 48 apples cost SOLUTION. If 16 apples cost 13 cents, 48 apples, which are 18, or 3 times 16 apples, cost 3 times 13 cents, or 39 cents. 11. If 15 oranges cost 14 cents, what will 60 oranges cost at the same rate? Ans. 56 cents. 12. What cost 125 sheep, if 25 sheep cost 137 dollars? Ans. $685. 13. If 56 cows cost $1000, what will 42 cows cost at the same rate? Ans. $750. 14. If 105 pens cost 77 cents, what will 225 pens cost at the same rate? 15. A has 20 cows, which is obtains of B's number, what equal B &? Ans. $1.65. of B's number; now if A part of A's number will Ans. of A's. CASE II. 186. To find the relation of a fraction to a num 5. 1 to 28? Ans. 26. 8. 30 10. A having $25, gave $5 to B and of the remainder to C; what part of $25 then remained? Ans. . 187. To find the relation of a number to a frac Ans. 414.5. 462 to 85? Ans. 2133.7. 63504 to 50%? Ans. 54. Ans. 154. Ans. 1260. as many, 4. 1896 to 24? 8. Mary had 12 minus of a quart; how many times Sarah's number equals Mary's? CASE IV. Ans. 1. 188. To find the relation of a fraction to a frac tion. 1. What part of 14 is? SOLUTION. is of 13, and 1, or one, is 15 times 7 times of 1, which equals 1 of 1, or §. equals 1 of 13. If 1 equals 1 of 13, equals of 1, and |