and employed 30 men to perform the labor. At the end of 40 days it was only half finished. How many additional laborers was he obliged to employ to perform the work within the time agreed upon? 47. A person, being asked the time of day, replied that it was past noon, and that of the time past noon was equal to of the time to midnight. What was the time? 48. A gentleman wishes his son to have $3000 when he is 21 years of age. What sum must be deposited at the son's birth, in a savings bank, which pays compound interest at the annual rate of 6%, so that the deposit shall amount to that sum when the boy becomes of age? 49. A note for $100 was due on Sept. 1st, but on Aug. 11th the maker proposed to pay as much in advance as will allow him 2 mo. after Sept. 1st to pay the balance. How much must be paid Aug. 11th, money being worth 6%? 50. What sum must a person save annually, commencing at 21 years may be worth $25000 when he is 40 years old, if he gets 6% compound interest on his money? 51. If a merchant sells of an article for what of it cost, what is his gain per cent. ? 52. If goods are sold so that of the cost is received for half the quantity of goods, what is the gain per cent. ? 53. A man sold a horse and carriage for $597, gaining by the sale 25% on the cost of the horse and 10% on the cost of the carriage. If of the cost of the horse equaled of the cost of the carriage, what was the cost of each? 54. If 300 cats can kill 300 rats in 300 minutes, how many cats can kill 100 rats in 100 minutes ? 55. A party of 8 hired a coach. If there had been 4 more the expense would have been reduced $1 for each person. How much was paid for the coach ? 56. I sold goods at a gain of 20%. If they had cost me of age, so that he $250 more than they did, I would have lost 20% by the sale. How much did the goods cost me? 57. A laborer agreed to work for $1.25 per day and his board, paying $.50 per day for his board when he was idle. At the end of 25 days he received $19. How many days was he idle ? 58. A is 20 years of age; B's age is equal to A's and half of C's; and C's is equal to A's and B's together. What is the age of each? 59. A and B were partners in a profitable enterprise. A put in $4500 capital and received of the profits. What was B's capital? 60. A man spent $4 more than half his money traveling, one-half what he had left and $2 more for a coat, $6 more than half the remainder for other clothing, and had $2 left. How much money had he at first? 61. A boy bought at one time 5 apples and 6 pears for 28 cents, and at another time 6 apples and 3 pears for 21 cents. What was the cost of each kind of fruit ? 62. A and B can do a piece of work in 20 days. If A does e as much as B, in how many days can each do it? 63. A man bought a farm for $5000, agreeing to pay principal and interest in 5 equal annual installments. What will be the annual payment, including interest at 6%? . 64. A carriage maker sold two carriages for $300 each. On one he gained 25%; on the other he lost 25%. Did he gain or lose by the sale? How much, and how much per cent. ? 65. If a ladder placed 8 feet from the base of a building 40 feet high, just reached the top, how far must it be placed from the base of the building that it may reach a point 10 feet from the top? 66. Mr. A. is 35 years of age and his son is 10. How soon will the son be one-half the age of the father? 67. A person in purchasing sugar found that if he bought sugar at 11 cents he would lack 30 cents of having money enough to pay for it, so he bought sugar at 102 cents and had 15 cents left. How many pounds did he buy? 68. A farmer had his sheep in three fields. of the number in the first field was equal to 1 of the number in the second field, and of the number in the second field was of the number in the third field. If the entire number was 434, how many were there in each field ? 69. A and B can do a piece of work in 10 days, B and C can do it in 12 days, and A and C in 15 days. How long will it take each to do it? 70. A, B and C pasture an equal number of cattle upon a field of which A and B are the owners-A of 9 acres and B of 15 acres. If C pays $24 for his pasturage, how much should A and B each receive? 71. How many acres are there in a square tract of land containing as many acres as there are boards in the fence inclosing it, if the boards are 11 feet long and the fence is 4 boards high? 72. What is the greatest number which will divide 27, 48, 90, and 174, and leave the same remainder in each case ? 73. A and B invested equal sums in business. A gained a sum equal to 25% of his stock, and B lost $225. A's money at this time was double that of B's. What amount did each invest? 74. A man at his marriage agreed that if at his death he should leave only a daughter, his wife should have of his estate; and if he should leave only a son, she should have He left a son and a daughter. What fractional part of the estate should each receive, and how much was each one's portion, if the estate was worth $6591? TEST QUESTIONS. 612. Define a unit; a number. Explain the necessity for a uniform system of grouping objects. In how many ways may numbers be represented? Name them. Define numeration; notation ; Arabic notation; Roman notation. Give the first principle of Arabic notation. Illustrate it. What is meant by “units of first order," etc.? Give the general principles of Arabic notation. What is meant by a period of figures? Give the names of the first seven periods. Give the rule for notation; for numeration. State how cents and mills are written in notation of U. S. money. What characters are employed in Roman notation? Give the principles of Roman notation. Define addition; sum, or amount; equation; like numbers. Describe the sign of addition; the sign of equality. How many cases are there in addition? Show the truth of the principles of addition. Repeat the rule for addition. Why do we begin at the right to add? Why are the numbers of the same order written in the same column? Define subtraction; minuend; subtrahend; remainder; difference. What is the sign of subtraction? What is it called? State the principles of subtraction. Show that they are true. Explain what is to be done when some figure of the subtrahend expresses more than the corresponding figure of the minuend. Define multiplication; multiplicand; multiplier; product; factors of the multiplier; abstract number. Describe the sign of multiplication. Give the principles of multiplication. Show that they are true. Show that multiplication is a special case of addition. Repeat the rule for multiplication. What steps in the process are for convenience? How may you multiply when there are ciphers on the right of either or both factors? Define division; dividend; divisor; quotient; remainder. What is the sign of division? In how many ways is division indicated? State the principles of division. Show that they are true. Show that division is a special case of subtraction. In how many ways may the remainder be expressed ? Illustrate each way by an example. What is a fraction? What is meant by long division? What is meant by short division? Which should precede the other? Why? What steps in the process of division are for convenience? What are necessary? How may you proceed when there are ciphers on the onel right of either divisor or dividend ? State the principles governing the relation of dividend, divisor, and quotient. Illustrate each by an example. Define analysis. Illustrate the process. Describe the parenthesis and vinculum, and show their uses. Define and illustrate what is meant by an integer; exact divisor; factor; a prime number; a composite number; an even number; an odd number. Give eleven facts relating to exact divisibility of numbers. Illustrate each statement by an appropriate example. What is meant by factoring? Prime factors? What is an exponent? State the principles relating to the prime factors of numbers. Illustrate the truth of these principles by appropriate examples. Give the rule for finding the prime factors of a number. Explain the process of multiplying by factors. Show the use of this process. Show how to divide by factors. Explain how to find the true remainder in division by factors. Give the rule for dividing by the factors of a number. What is meant by cancellation? Upon what principle is the process based? Illustrate the process. Define what is meant by a common divisor; the greatest common divisor; numbers that are prime to each other. What is the principle underlying the greatest common divisor? Give the ordinary method of finding the greatest common divisor when the numbers are small. Solve an example, and give the analysis when the numbers can not be readily factored. What is a multiple? Define what is meant by a common multiple; the least common multiple. State the principle upon which the processes in least common multiple are based. Solve an example showing the truth of the principle. Define and illustrate what is meant by the terms fraction; unit of a fraction; fractional unit; the denominator; the numerator; the terms of a fraction; a proper fraction; an improper fraction; a mixed number; a common fraction; a decimal fraction. How are fractional expressions read? Interpret the expression .. What is meant by reduction of fractions? What is Case I? When is a fraction reduced to larger or higher terms? Upon what principle does the process in Case I depend? What is Case II? What is meant by reducing a fraction to smaller or lower terms? To smallest or lowest terms? Upon what principle is the process in Case II based? What is Case III in reduction? Solve an example illustrating the process. What is Case IV? Solve an example illustrating |