145. A person bought a number of horses for $1404. If there had been 3 less, each would have cost him $39 What was the number of horses and the cost of more. each? 146. Find a number of four figures which increase from left to right by a common difference 2, while the product of these figures is 384. Ans. 2468. 147. A rectangular garden 24 rods in length and 16 in breadth is surrounded by a walk of uniform breadth which contains 3996 square feet. What is the breadth of the walk? Ans. 3 feet. 148. A square field containing 144 ares has just within its borders a ditch of uniform breadth running entirely round the field and covering 381.44 centares of the area. What is the breadth of the ditch? Ans. 0.8 meter. 149. A and B hired a pasture into which A put 5 horses, and B as many as cost him $5.50 a week. If B had put in 4 more horses, he ought to have paid $6 a week. What was the price of the pasture a week? Ans. $8. 150. A father dying left $3294 to be divided equally among his children. Had there been 3 children less, each would have received $183 more. How many children were there? 151. A merchant bought a quantity of tea for $66. If he had invested the same sum in coffee at a price $0.77 less a pound, he would have received 140 pounds more. How many pounds of tea did he buy? 152. Find two quantities such that their sum, product, and the sum of their squares shall be equal to one another. Ans. (3 ±√ — 3) and 1 (3 ‡ √ — 3). 153. Find two numbers such that their product shall be 6, and the sum of their squares 13. 154. A and B talking of their ages find that the square of A's age plus twice the product of the ages of both is 3864; and four times this product, minus the square of B's age, is 3575. What is the age of each? Ans. A's, 42; B's, 25. 155. Find two numbers such that five times the square of the less minus the square of the greater shall be 20; and five times their product minus twice the square of the greater shall be 25. 156. A and B purchased a wood-lot containing 600 acres, each agreeing to pay $17500. Before paying for the lot, A offered to pay $20 an acre more than B, if B would consent to a division and give A his choice of situation. How many acres should each receive, and at what price an acre? Ans. A, 250 acres at $70 an acre; B, 350 at $50. 157. A merchant bought two pieces of cloth for $175. For the first piece he paid as many dollars a yard as there were yards in both pieces; for the second, as many dollars a yard as there were yards in the first more than in the second; and the first piece cost six times as much as the second. What was the number of yards in each piece? Ans. In 1st, 10 yards; in 2d, 5. 158. Two sums of money amounting to $14300 were lent at such a rate of interest that the income from each was the same. But if the first part had been at the same rate as the second, the income from it would have been $532.90; and if the second part had been at the same rate as the first, the income from it would have been $490. What was the rate of interest of each? Ans. First, 7 per cent; second, 7 per cent. 159. Divide 29 into two such parts that their product will be to the sum of their squares as 198: 445. 160. What is the length and breadth of a rectangular field whose perimeter is 10 rods greater than a square field whose side is 50 rods, while its area is 250 square rods less than the area of the square field? Ans. Length, 75 rods; breadth, 30. 161. A rectangular piece of land was sold for $5 for every rod in its perimeter. If the same area had been in the form of a square, and sold in the same way, it would have brought $90 less; and a square field of the same perimeter would have contained 272 square rods What were the length and breadth of the field? Ans. Length, 49; breadth, 16 rods. more. 162. A starts from Springfield to Boston at the same time that B starts from Boston to Springfield. When they met, A had travelled 30 miles more than B, having gone as far in 12 days as B had during the whole time; and at the same rate as before B would reach Springfield in 5 days. How far from Boston did they meet? Ans. 42 miles. 163. The product of two numbers is 90; and the dif ference of their cubes is to the cube of their difference as 13: 3. What are the numbers? 164. A and B start together from the same place and travel in the same direction. A travels the first day 25 kilometers, the second 22, and so on, travelling each day 3 kilometers less than on the preceding day, while B travels 14 kilometers each day. In what time will the two be together again? Ans. 8 days. 165. A starts from a certain point and travels 5 miles the first day, 7 the second, and so on, travelling each day 2 miles more than on the preceding day. B starts from the same point 3 days later and follows A at the rate of 20 miles a day. If they keep on in the same line, when will they be together? Ans. 3 or 7 days after B starts. 166. A gentleman offered his daughter on the day of her marriage $1000; or $1 on that day, $2 on the next, $3 on the next, and so on, for 60 days. The lady chose the first offer. How much did she gain, or lose, by her choice? 167. The arithmetical mean of two numbers exceeds the geometrical mean by 2; and their product divided by their sum is 3. What are the numbers? 168. A father divided $130 among his four children in arithmetical progression. If he had given the eldest $25 more and the youngest but one $5 less, their shares would have been in geometrical progression. What was the share of each? 169. The sum of the squares plus the product of two numbers is 133; and twice the arithmetical mean plus the geometrical mean is 19. What are the numbers? 170. The sum of three numbers in geometrical progression is 117; and the difference of the second and third minus the difference of the first and second is 36. What are the numbers? 171. There are four numbers in geometrical progression, and the sum of the second and fourth is 60; and the sum of the extremes is to the sum of the means as 7:3. What are the numbers? SECTION XXV. LOGARITHMS. 241. LOGARITHMS are exponents of the powers of some number which is taken as a base. In the tables of logarithms in common use the number 10 is taken as the base, and all numbers are considered as powers of 10. By Arts. 119, 120, 10° 1, that is, the logarithm of 1 is 0 = Therefore, the logarithm of any number between 1 and 10 is between 0 and 1, that is, is a fraction; the logarithm of any number between 10 and 100 is between 1 and 2, that is, is 1 plus a fraction; and the logarithm of any number between 100 and 1000 is 2 plus a fraction; and so on. Therefore, the logarithm of any number between 1 and 0.1 is between 0 and -1, that is, is -1 plus a fraction; the logarithm of any number between 0.1 and 0.01 is between -1 and 2, that is, is 2 plus a fraction; and so on. The logarithm of a number, therefore, is either an integer (which may be 0) positive or negative, or an integer positive or negative and a fraction, which is always positive. |