of 30 miles an hour, the other at 35 miles an hour. In how many hours will they be 260 miles apart? 35 m.p.h. -30 m.p.h. 23. Two trains start at the same time from different stations 360 miles apart. One travels at the rate of 40 miles an hour and the other at the rate of 32 miles an hour. How far does the slower train travel before meeting the faster train? 24. Two hours after a train left a certain station, a second train started out and overtook the first train in 5 hours. To do this, the second train had to run 10 miles an hour faster than the first train. What was the speed of each? 25. An excursion boat makes the up-river trip at 8 miles per hour and the down-river trip at 12 miles per hour, the whole time being 10 hours. Find the distance. 26. A tank holding 7200 gallons has three pipes. The first lets in 6 gallons per minute, the second 8 gallons, and the third 10 gallons. In how many minutes will the tank be filled? 27. The front and rear wheels of a carriage are 12 and 14 feet respectively in circumference. How many feet will the carriage have passed over when the front wheel has made 200 revolutions more than the rear wheel? 28. Twenty persons rented a hall, but four being unable to pay, each of the others had to pay $1 more than his share. Find the rental of the hall. 29. A stream runs at the rate of 2 miles an hour. A man swims a certain distance up the river in 3 hours and the same distance down in 1 hr. Find his rate of swimming in still water. 30. A man having only half dollars and quarters wishes to give some persons 75 cents each, but found that he had not money enough by $1.25. He therefore gave them 50 cents each and had $1.50 left. How many persons were there? 31. A grocer has two kinds of coffee, one worth 40 ¢ a pound and the other 30 é a pound. He makes a mixture from these of 100 pounds, worth 32 ¢ a pound. How many pounds of each kind does he take? 32. A sum of money is divided among three persons, A, B, and C, so that A and B together have $10, A and C $12, and B and C $14. How much has each? 33. A merchant sold 3 equal lots of cloth for $56. For the first lot he received $2 per yard, for the second $2.75, for the third $3.25. How many yards of each kind did he sell? 34. A tree 70 feet high was broken so that the part broken off was four times the length of the part left standing. Find the length of each part. 35. Twice the excess of a certain number over 5 is equal to the number increased by 5. Find the number. 36. Twenty-one times a number exceeds 75 by as much as four times the number is less than 75. Find the number. 37. A has $20.25 and B has $3.75. How much must A give to B in order that he may have just three times as much as B? 2. A man paid yearly a certain amount for taxes and twice as much for improvements and received for rent three times as much as he paid for improvements. If his net gain per year was $300, what were his taxes? 3. A speculator who doubled his money by a fortunate investment, afterward lost $600, but he still had $400 more than the original sum. How much did he invest? 4. Solve and verify: 7(3x-6) + 5(x − 3) + 4(17 — x) = 77. 5. A's age exceeds B's by 14 years. Eight years ago A was 3 times as old as B. Find the present age of each. 6. Divide $135 among A, B, and C, so that A's share is 3 times B's and B's is twice C's. 7. Solve and verify, 2(x + 1)-3(x + 1) + 9(x+1)+ 7(x + 1). 18 = 8. A man has $4.75 in quarters and dimes, and he has 5 more quarters than dimes. How many coins of each kind has he? 9. A teacher proposes 16 problems to a pupil. The latter is to receive 5 marks in his favor for each problem solved, and 3 marks against him for each problem not solved. If the marks in his favor exceed those against him by 32, how many problems will he have solved? 10. A man leaves his estate amounting to $7500, to be divided among his wife, two sons, and three daughters. A son was to have twice as much as a daughter, and the wife $500 more than all the children together. Find the share of each. 11. A man and two helpers together earned $7.50 a day. How much did each earn per day if the man earned 4 times as much as each helper? 12. Solve and verify: 6x + 4(4x+2)+3(2x+7) = 85. 13. A bicyclist averaging 12 miles an hour is 52 miles ahead of an automobile running 20 miles an hour. How soon will the automobile overtake him? 14. A rectangular shed sheltering an airship was 272 feet in perimeter. If twice its length was 26 feet more than four times its width, find its dimensions. 15. Forty stamps, two's and three's, cost 95 ¢. How many were there of each? 16. A dealer paid $185 for some boxes of candles, at $9 for some, and at $6.50 for the others. The number of the latter was 2 less than twice the number of the former. How many of each kind did he buy? 17. Divide $152 among 5 men, 7 women, and 30 children, giving to each man $4 more than to each woman, and to each woman three times as much as to each 18. A man went to a bank with a check for $36 and asked to have it cashed in half dollars, quarters, dimes, and nickels, of each the same number. What was the number? 19. Solve: 5(3x-5) - 2(3x-5)= 44+ 8(5 - 3x). 20. Solve and verify: 80 6(4x + 3) = 7x − 3 (6x + 1). CHAPTER IV ADDITION AND SUBTRACTION 27. Algebraic Addition and Subtraction differ from the similar operations in Arithmetic only in that algebraic as well as arithmetical numbers are included, that is, negative, as well as positive, numbers. To indicate the addition or subtraction of algebraic numbers, a single number is often inclosed in a parenthesis. Thus 4 + ( −3) indicates that −3 is to be added to 4, the parenthesis being used to avoid confusion between the sign of operation and the minus sign belonging to 3. When numbers have the same sign, they are said to have like signs, and when their signs are different, they have unlike signs. It is evident that when two numbers have like signs, their sum is found by simple arithmetical addition, giving to the sum, the sign of the numbers. But, since positive and negative numbers are opposite in kind, they will oppose each other when combined by addition, that is, the addition of a negative number will have the effect of diminishing the positive number. b −a +(+b) = = -a +(-b) -a We therefore have the following Rule for Adding Two Algebraic Numbers. 1. If the numbers have like signs, find the sum of their |