49. A person invests $10,000 in three per cent bonds, $16,500 in three and one-half per cents, and has an income from both investments of $1056.25. If his investments had been $2750 more in the three per cents, and less in the three and one-half per cents, his income would have been 62 cents greater. What price was paid for each class of bonds?

50. The sum of $2500 was divided into two unequal parts and invested, the smaller part at two per cent more than the larger. The rate of interest on the larger sum was afterwards increased by 1, and that of the smaller sum diminished by 1; and thus the interest of the whole was increased by one-fourth of its value. If the interest of the larger sum had been so increased, and no change been made in the interest of the smaller sum, the interest of the whole would have been increased one-third of its value. Find the sums invested, and the rate per cent of each.

NOTE VII. If x represent the number of linear units in the length, and y in the width, of a rectangle, xy will represent the number of its units of surface; the surface unit having the same name as the linear unit of its sides.

51. If the sides of a rectangular field were each increased

by 2 yards, the area would be increased by 220 square yards; if the length were increased and the breadth were diminished each by 5 yards, the area would be diminished by 185 square yards. What is its area?

52. If a given rectangular floor had been 3 feet longer and 2 feet broader it would have contained 64 square feet more; but if it had been 2 feet longer and 3 feet broader it would have contained 68 square feet more. Find the length and breadth of the floor.

53. In a certain rectangular garden there is a strawberrybed whose sides are one-third of the lengths of the corresponding sides of the garden. The perimeter of the garden exceeds that of the bed by 200 yards; and if the greater side of the garden be increased by 3, and the other by 5 yards, the garden will be enlarged by 645 square yards. Find the length and breadth of the garden.

NOTE VIII. Care must be taken to express the conditions of a problem with reference to the same principal unit.

Ex. In a mile race A gives B a start of 20 yards and beats him by 30 seconds. At the second trial A gives Ba start of 32 seconds and beats him by 9

Find the rate per hour at which each runs.

Let x number of yards A runs a second,

and y = number of yards B runs a second.
Since there are 1760 yards in a mile,

yards.

number of seconds it takes A to run a mile,

number of seconds B was running in the first and second trials, respectively.

= 30,

1

513
That is, A runs
or of a mile in one second;
1760' 300'

and in one hour, or 3600 seconds, runs 12 miles.

Likewise, B runs 10,95 miles in one hour.

54. In a mile race A gives B a start of 100 yards and beats him by 15 seconds. In the second trial A gives B a start of 45 seconds and is beaten by 22 yards. Find the rate of each in miles per hour.

55. In a mile race A gives B a start of 44 yards and beats him by 51 seconds. In the second trial A gives B a

start of 1 minute and 15 seconds and is beaten by 88 yards. Find the rate of each in miles per hour.

56. The time which an express-train takes to go 120 miles is of the time taken by an accommodation-train. The slower train loses as much time in stopping at different stations as it would take to travel 20 miles without stopping; the express-train loses only half as much time by stopping as the accommodationtrain, and travels 15 miles an hour faster. Find the rate of each train in miles per hour.

57. A train moves from P towards Q, and an hour later a second train starts from Q and moves towards P at a rate of 10 miles an hour more than the first train; the trains meet half-way between P and Q. If the train from P had started an hour after the train from Q its rate must have been increased by 28 miles in order that the trains should meet at the half-way point. Find the distance from P to Q. 58. A passenger-train, after travelling an hour, meets with an accident which detains it one-half an hour; after which it proceeds at four-fifths of its usual rate, and arrives an hour and a quarter late. If the accident had happened 30 miles farther on, the train would have been only an hour late. Determine the usual rate of the train.

59. A passenger-train after travelling an hour is detained 15 minutes; after which it proceeds at three-fourths of its former rate, and arrives 24 minutes late. If the detention had taken place 5 miles farther on, the train would have been only 21 minutes late. Determine the usual rate of the train.

60. A man bought 10 oxen, 120 sheep, and 46 lambs. The cost of 3 sheep was equal to that of 5 lambs; an ox, a sheep, and a lamb together cost a number of dollars less by 57 than the whole number of animals bought; and the whole sum spent was $2341.50. Find the price of an ox, a sheep, and a lamb, respectively.

61. A farmer sold 100 head of stock, consisting of horses, oxen, and sheep, so that the whole realized $11.75 a head; while a horse, an ox, and a sheep were sold for $110, $62.50, and $7.50, respectively. Had he sold one-fourth of the number of oxen that he did, and 25 more sheep, he would have received the same sum. Find the number of horses, oxen, and sheep, respectively, which were sold.

62. A, B, and C together subscribed $100. If A's subscription had been one-tenth less, and B's one-tenth more, C's must have been increased by $2 to make up the sum; but if A's had been one-eighth more, and B's one-eighth less, C's subscription would have been $17.50. What did each subscribe?

63. A gives to B and C as much as each of them has; B gives to A and C as much as each of them then has; and C gives to A and B as much as each of them then has. In the end each of them has $6. How much had each at first?

64. A pays to B and C as much as each of them has; B

pays to A and C one-half as much as each of them then has; and C pays to A and B one-third of what each of them then has. In the end A finds that he has $1.50, B $4.16, C$.58. How much had each at

CHAPTER XIII.

INVOLUTION AND EVOLUTION.

198. The operation of raising an expression to any required power is called Involution.

Every case of involution is merely an example of multiplication, in which the factors are equal.

(2a)=2a3× 2a3=4ao.

Thus,

199. Any power of a power of a number is obtained by taking the product of the exponents of the powers. The proof of this law of exponents, in its general form, is:

(am)n = am xam xam x to n factors,

= a

‚m + m + m +..... to n terms,

mn

Hence, if the exponent of the required power be a composite number, it may be resolved into prime factors, the power denoted by one of these factors may be found, and the result raised to a power denoted by another, and so on. Thus, the fourth power may be obtained by taking the second power of the second power; the sixth by taking the second power of the third power; the eighth by taking the second power of the second power of the second power.

200. From the Law of Signs in multiplication it is evident that,

I. All even powers of a number are positive.

II. All odd powers of a number have the same sign as

the number itself.