100. A person took $6 with him to distribute equally among some poor persons; but, as 5 of them were absent, the remainder received 10 cents apiece more than they otherwise would have received. How many poor persons were there? 101. A merchant bought a piece of cloth for $40, and after cutting off 4 yards, sold the remainder at $2.50 per yard, and received what the whole cost him. How many yards were there, and what did they cost? Ans. 20 yards, at $2. 102. Find the greatest common divisor of a5-5a*x+10a3x2 - 10a2x2+5ax* — x5 and a3 +x3 — ax2 — a2x. Ans. a2-2ax+x2. but if 103. A broker bought two cabinets at a sale, having been told that a $10 gold piece was hidden in one of the two. If in the one, he thought it should be worth twice the other; in the other, it ought to be worth three times the first. value did he put upon the cabinets without the gold piece? 104. Multiply aa+b+c a by a−24+c". What 105. Two boats were sent out from a ship on a whaling excursion, with crews in the ratio of 3 to 5. Meeting some time after, they considered it would be better to divide the hands equally between the two boats, and did so by removing 6 men from one boat to the other. What was the whole number of men sent out? Ans. 48. 106. Find the value of x in the equation √/x+α = 2√/x+b. 107. A party of laborers were sent to remove a bank of earth containing 350 cubic yards; but just as they were commencing, four of them were disabled by an accident, in consequence of which each of the rest had ten additional yards of earth to remove. What was the number of the party? Ans. 14. 108. A traveller, having proceeded 175 miles by railway, complained of the slowness of the train, and showed that, if it had only gone five miles per hour faster, he would have accomplished his journey in less time by an hour and three-quarters. What was the rate at which the train went? Ans. 20 miles per hour. 109. Find the least common multiple of (x+2a)3, (x-2)3 and x2+4a2. 3 110. Find two numbers such that, if 6 be added to each, they shall be to each other as 4 to 5, and if 4 be taken from each, they shall be to each other as 2 to 3. Ans. 14; 19. c, to find x, y and z. Ans. x=4(3±√/5). 113. A gentleman bought containing together 40 acres. lars per acre as there were acres in the field, and the prices of the two were in the proportion of 4 to 9. What were the areas of the two fields? separately two contiguous fields Each of them cost as many dol 114. A person rents a certain number of acres of pasture land for $70. He keeps 8 acres in his own possession, sublets the remainder at 25 cents per acre more than he gave, and thus covers his rent and $2 over. What was the number of acres? 115. A and B are two towns situated 18 miles apart on the same bank of a river. A man goes from A to B in 4 hours, by rowing the first half of the distance and walking the second half. In returning, he walks the first half at the same rate as before, but the stream being with him, he rows 1 miles per hour more than in going, and accomplishes the whole distance in 3 hours. Find the rates of walking and rowing. Ans. 4 miles per hour walking; 4, rowing at first. 116. Reduce the surds /300 and 1/75 to the same radical, and find their sum. 117. Find two numbers in the ratio of 5 to 6, such that their sum has to the difference of their squares the ratio of 1 to 7. Ans. 35; 42. x 21 23 118. Solve the quadratic =+ 119. An orange peddler bought a number of oranges at the rate of 5 for 2 cents. He then arranged the good and bad in two separate baskets, containing equal numbers, and sold the one basketful at 3 for 1 cent, and the other at 3 for 2 cents. In selling them, he was told he would make no profit upon them; but when he had sold the whole he found he had gained 6 cents. How many oranges did he buy and sell? Ans. 360. 120. Divide 4x-12 by √x−3. Ans. 8√x-3. 121. A and B engage in partnership with a capital of $5000; A has his money in for 3 months, and B for 2 months, and each at last realizes $4950 of capital and profit. What was the original contribution of each? Ans. A, $2250; B, $2750. 122. The seventh term of a geometrical progression is 1000, and the common rate is 104. What is the first term? Ans. 790.31+ 123. A certain rectangle contains 300 square feet; a second rectangle is 8 feet shorter and 10 feet broader, and also contains 300 square feet; what is the length and breadth of the first rectangle? Ans. 20 feet; 15 feet. 124. What two numbers are such that their product is 45, and the difference of their squares is to the square of their difference as 7 is to 2? Ans. 9 and 5. 125. Two detachments of infantry being ordered to a station 39 miles from their present quarters, begin their march at the same time; but one party, by marching of a mile per hour faster than the other, arrive there an hour sooner. Required their rates of marching. Ans. 34 and 3 miles per hour. APPENDIX. SECTION XLVII. GENERALIZATION. 364. A General Problem is one in which all the quantities are represented by letters. The result obtained by the solution of a general problem expresses the value of the unknown in the terms of the known quantities. 365. Generalization is the process of solving a general problem, and interpreting the resulting expression. The formulas, or general expressions, derived from the solution of general problems, when interpreted, form rules for the solution of all similar problems. 366. A problem is generalized when letters are made to represent its known quantities. EXERCISES. 367.-Ex. 1. The sum of two numbers is s, and their difference is d. What are the two numbers? SOLUTION. Let x=the greater number, and y = the less; then, by the conditions, x+y=s, and x-y=d. y = the less. x+y=8 x-y=d = the greater; (1) (2) Adding equations (2) and (1), and subtracting (2) from (1), we obtain equa tions (3) and (4). Whence, x= quantities whatever; hence, the values of x and y are general, and 225 equations (5) and (6) are formulas for finding two numbers when their sum and difference are given. Hence, the following, 368. Rule for finding two numbers from their sum and difference. Add the difference to the sum, and divide by 2, for the greater of the two numbers; subtract the difference from the sum, and divide by 2, for the less of the two numbers. 1. The sum of two numbers is 391, and their difference is 53. What is the greater number? 2. A and B hire a pasture together for $65, and A is to pay $13 more than B. How much is each to pay? Ans. A, $39; B, $26. 3. In an election, the aggregate of votes for A and B was 9637, and B's majority over A was 593. How many votes did each receive? 369.-Ex. 1. Divide the sum S among A, B and C, in the proportion of the numbers m, n and p. |